This is probably more of a loophole than the answer you are looking for, but an electronic circuit with a quartz oscillator need not be programmable. It doesn't even need an electronic memory. You'd need to add some frequency dividers, which can also be analog, to provide the final output signal that controls the rocket.

Here's another thought: ultra-precise timing of the burn is only needed if your rocket is on/off only. If you can throttle it back, then you can do the main acceleration/deceleration burn at full power, followed by an orbital correction at much lower power. Or you can use a different set of engines for that part of the manoeuvre.

Addendum

Zubrin's original article says:

In order to be able to analyze the NSWR completely, the
requirement is a computer code which solves the coupled multigroup
neutron transport equation, hydrodynamic equation of
motion, and heat transfer equation as a single system.

So, developing this system without computers is a no-no. However, <Jedi hand gesture> this is not the plot hole you're looking for. In any case, throttling a NSWR seems to be impossible, since the geometry of the plenum chamber must be tuned to the propellant velocity, concentration and yield, ultimately thrust, while remaining strong enough to support a continuous nuclear explosion. You just can't make it telescoping.

The answer then is two engines. The NSWR primaries fire for a fixed amount of time, and and push the vehicle from planetary orbit to transfer The amount of propellant you've used is on the order of 10% of ship's weight. Keep in mind that it might be easier to precisely control the amount of propellant used than the time of the burn.

Then you coast for a few days, carefully observe your trajectory and work out the necessary course corrections. Without computers you give yourself plenty of time and double-check your decimal points! Then fire up your low thrust ion engines an keep them on for hours or days, until you are on track.

They could use pulsars emission as clock

Neutron stars are very dense, and have short, regular rotational periods. This produces a very precise interval between pulses that ranges from milliseconds to seconds for an individual pulsar. [...] Certain types of pulsars rival atomic clocks in their accuracy in keeping time.

A suitable observatory has to be pointed to one or more pulsars and the collected EM emission is converted to an output electric signal (think something like a photodiode), and by collecting and measuring their emission act as "pace maker" for the ship. By measuring the interval dilation for a pulsar, one can also account for relativistic deviations.

Since rotation is used to stabilize objects in space, the ships could be put in rotation with the rotation axis pointing at a pulsar, providing also an indirect way to measure proper alignment (when signal is lost orientation has been disrupeted)

Objective time is irrelevant, only subjective time matters

All your navigation calculations can be done on subjective time. All your shift patterns are going to work on subjective time.

You're far better off building a really good shipboard clock and doing good calculations relative to that, than attempting to automatically convert from some sort of external time prompt. Once you're out of contact with others, the only time that matters is ship time. So for these purposes, we're not going to correct for relativistic effects at all.

So what options are there?

For a traditional no computers environment, we need a good old fashioned mechanical clock. We can cheat and put an electric motor in the back so you don't need to worry about winding it, but we're better off rigging the motor against some nice constant force springs. That way rather than powering the clock itself the motor powers the winding mechanism. Of course this isn't going to be the most accurate option, falling at around 99.9977%

For a more accurate modern approach you want quartz. Something like the Casio F91W, the terrorists favourite bomb trigger due to low cost and legendary accuracy (99.9998%). Again it's a tried and tested system, the accuracy and time loss is a known quantity that can be calculated against. (For a premium system you can get up to 99.9999% accuracy, but where's the fun in that?)

However the trouble with both quartz digital watches or something like an atomic clock is the computer memory, a digital clock is fundamentally a single function computer. And in either case to know when to trigger or end your burns you need to be able to program a time into it to either count up or down to.

However it's possible to build an analogue clock with a quartz timekeeper* giving you a mechanical memory and mechanical output triggers. This should create your "ideal" shipboard clock without upsetting any of the computer zealots. The key difference being the amount of margin for error you're required to put into your navigation calculations at every stage.

Why doesn't absolute time matter?

In the simplest case, because you're going from "here" to "there". When you get "there" you can pick up your new vectors from the local space traffic control. The time where you came from is irrelevant, only the relative position of where you're going next. Your navigation clock doesn't even need to be a clock, it only needs to be a timer. The only times that matter for navigation are the time to next burn and the burn duration.

But we're an exploration vessel, we don't go to ports!

Now it helps if you have a separate clock for your mission time, but it's still not strictly required. You know where you are, you know the location of this place relative to the other places on your route and the calculations for their relative movements. It can all be calculated, it's going to get harder for every extra step you take as the margins for error increase, but it can be done.

This can be made easier with something that will make the over 70s happy. Lookup tables. When you're at point A and visible object B is on vector AB, visible object C is on vector AC, then you can look up your next journey AD, by knowing the difference between AB and AC.

* I won't quote accuracy on these clocks for now as they tend to be at the very cheap end of the market. I'm sure they could be made accurate but there's no call for it.

I'm going to be a bit contrarian and say that any old mechanical pocket-watch will do. It is rather more important to know where you are and how fast you are going. (Which can be solved with radar, radio beacons and other WWII-technology) If you have a torch-ship, you don't need precise clocks at all. At the speeds you are going, we are talking about burn times of at least hours, if not permanent burns in the classic torch-ship style. Burning for a few seconds shorter or longer will not make much of a difference. In fact I guess that impurities in your fuel and slight imperfections in you combustion chamber would have a greater impact. Also, if you are not going quite at the velocity you want to be going, you can just do some adjustment burns, assuming that you left the time for them in your flight plan.

There is also not much need to worry about orbital elements, since the trajectory of a torch ship is really close to a straight line. You just aim at where you want to be going (say the point where Mars is going to be in two weeks), signal "full speed ahead" to your engine-room and then turn around at the midpoint. (Or rather a short bit before, just to be safe.) When you get close enough, you then simply start matching speed until you end up on a slow approach. Getting into a proper orbit in the end might be a bit more finicky, but again, if you have the fuel to spare, it's not terribly hard. See also this nice video on the topic. (And note that he indeed eyeballs the timings and orbital insertion)

All this could actually make the setting more interesting. While any officer who passed his interplanetary navigations class can safely fly a ship, an experienced captain could potentially shave a few hours off the travel time, or a young hotshot lieutenant could ignore the regulation safety margins and go for a suicide burn in the hope of impressing his superiors.

Allow only mechanical computers. There were difference engines (the first computers) that were purely mechanical.

In Abraham Lincoln's time, the census was performed on punch cards (male/female, age, region, etc.). Mechanical counters advanced each time the proper hole or set of holes were present.

There are also specially carved cams that can open and close valves at predefined, changing rates.

Also, as @Separatrix mentioned, lookup tables work as well.

Space travel isn't impossible without computers, it just takes longer.

You can't go directly from point A to point B. You go to a point near where B should be, stop, take your readings, correct your course and go to another point closer to B, rinse and repeat until you can make it in one more hop.

In this scenario, the better the charts and tables you have and the more skill the pilot has, the closer to B you can get without getting into trouble. This will bring back the ace pilot.

Standard charts and book learning will allow a pilot to make a safe trip from point A to point B but an ace pilot with top of the line charts and equipment (the cams are blueprinted) can shave days or weeks off of the journey because they will need fewer recalculations.

My mother's college dictionary defined computer as: "One who computes."

So break out your slide rules, it's time for some old school SF.

BTW, you should look into George O. Smith's The Complete Venus Equilateral (wiki). This is the author that invented the Star Trek transporters (and had them ruin the economy). Thrust was provided by vacuum tubes and all calculations were done by hand or through machined cams.

You might find the Ford Mark 1A Fire Control Computer an interesting case study, it was a mechanical machine used for gun laying on battleships (essentially very precise ballistics and targeting calculations, not a million miles from calculating burns) and while the outputs to the guns was via syncro resolver the computations were almost wholly mechanical. They were incredible machines, especially as a battleship is not the most pleasant environment for precision mechanisms.

Sines and cosines, exponents and logs can be done as rollers moving on profiled metal cams, integration can be done by wheels moving on rotating disks (Or by accumulating fluid in a tank), mechanical computing devices were very much a thing before electronic computers swept all before them.

Syncro resolvers need no memory and can couple the movements of these very precise mechanisms to massive motors for moving nozzles as easily as they can move the guns on the Iowa, and gyroscopes are essentially mechanical.

For time, two big quartz crystals in oven oscillators at say 100KHz and 100.1KHz or so, a mixer, lowpass filter, and power amplifier and I have an accurate 100Hz, feed that to a synchronous motor and a gear chain, followed by a cam switch, job done, and nothing programmable. IIRC this was how some early manned space flights were sequenced (Well not the beat frequency thing, but mechanical cam switches, IIRC Gagarins light used these).

A quartz derived clock done the obvious way suffers from needing electronic flipflops operating at some speed, which are essentially single bit memory cells, so that is out, but a diode ring mixer has no memory it is just a (not very good) analogue multiplier that takes two voltage signals and multiplies them together. At the output of the mixer you get sum and difference frequency terms for each of the input frequencies.

To build an accurate electro mechanical clock without memory you take two stable oscillators (exact frequency does not much matter but the difference must be stable) and apply the outputs to a mixer, at the output of which will be two signals, one being the sum of the individual oscilators frequencies and the other being the difference, you filter out the difference term at maybe 50 - 100Hz or so, drive a power amplifier and use the resulting stable ac to drive a synchronous motor/gearbox combination that can then mechanically drive your cam based sequencer as well as your clock displays.
Time was the clocks in peoples houses often worked this way with the stable frequency coming from the mains grid that went to some lengths to ensure that over the course of a day it averaged to the right number of cycles.

For navigation a decent sextant can get your position anywhere in the solar system if you have the time, and good enough calculations for burns should be well within the capability of mechanical machines (Your constraints said no electronics computers (And you defined computer as having a memory), you would be amazed by what people would do with a combination of mechanics, optics and electrics).

The planets themselves, and their moons, form their own clock. Jupiter's moons were in fact used historically, before the invention of accurate chronometers: http://www.oceannavigator.com/May-June-2004/Longitude-by-Jove-navigating-with-Jupiters-moons/

I’m gonna take a shot at this. I like the sextant idea, with one that is detailed enough to use in conjunction with planets and stars while in orbit. In addition, on board there can be a mechanical rotating map of the solar system. This map would need to be very detailed while it is capable of having all the moving orbits of planets and moons and asteroids etc. Each trajectory, from planet or moon to wherever, would need to follow a planned course that uses this mechanical orbiting map like a micrometer. And uses basic trigonometric equations along with a sundial or shadow ruler, where the triangular shadows upon such a map would indicate the next thrust by distances.

Another idea to add would be the use of lenses. These lenses or glass spheres could better detail the pinpoints of light from the distant sun as it moves across such a map. But of course this would require a trajectory that was in constant light.

Rereading the question, One could add an electromechanical control simply by using an engine and a transmission, if not simply as a car would. If the gauges were read as revolutions per minute or hour as in comparison to km/s, once the parameter or distance was met the gear would shift and engage any switch. Sounds weird and inelegant but it would work as a clock, just needs a pilot and some type of fuel or electricity.

I will also add that since “nuclear” is already being used, and this is a thousand or more years in the future, perhaps the properties of nuclear decay have been better mapped and can provide some type of new time system beyond temperature.

In conclusion I guess the most accurate system we have for time is by orbits and revolutions from planets etc in space, because time is relative to what we believe it to be in relation to those things. All time changes based on the calculations of days and years and seasons, not by any concrete system that could be applied in space. So as the comouter makes calculations and changes, a mechanical clock would need to be very complicated if not infinite in its capabilities of reading how time changes in these different places (orbits, distances from sun, perhaps even distance from center of galaxy)because we only know theoretically what time does there.

The Atomic Clock Works

•The 'best' clock means the most accurate

Atomic clocks are among the most accurate and precise we have. Their numbers are obtained by measuring the state of an atom.

•The clock must be able to be accurately corrected for relativistic effects. 100 km/s isn't that fast, but errors can add up over time.

For the purpose of ship based operations, which this scenario expressly details, relativity is irrelevant. All ship based functions are relative to the ship as is time thus it isn't necessary to account for relativity.

Furthermore, the only times that relativity might be needed to account for is when defining offsets from ship time to port time. This can be done mathematically at the destination when it becomes relevant.

•The clock must give an analog signal output that can be integrated into other electro-mechanical devices. For example, the only way for an old-timey pocket watch to control a burn time is through a human operating a switch.

An atomic clock can be constructed in such a way that it is entirely analog containing no digital memory as per the scenario rules. How you want the signal can be as contrived as you want.

As for the scenario, the original Apollo Missions to the moon had as much digital automation as a pocket watch. A smart phone today is literally smarter than the lunar lander. This genre that you are describing is very reminiscent of old sci-fi literature like Heinlein with his constant 'Astrogators' profession (people who literally hand did all the astrophysical math).

I think I'm going to answer this from a slightly different perspective: you've got the wrong mentality for what spacefaring would be like.

You're picturing a modern-style of "Tracking your position, using accurate time-keeping to determine position, etc." Problem is, like a lot of people have mentioned, doing that sort of thing without a computer in space-travel is pretty much impossible.

But what if you alter your perception of what space travel is like? The Space Program is notoriously "tight" - you have to save every watt you can, you have to reduce mass as much as possible, you have to find optimal trajectories, you have to precisely chart the landing lest you skip of the atmosphere or burn up coming in too steep. It's probably the most efficiency-demanding field there is - which is why computers are a MUST.

But what if your society doesn't have those limitations? Getting into orbit isn't tough - you've got engines that don't require nearly as much weight in fuel. You've got hardened materials that make orbital approaches more forgiving (along with engines that you can afford to use to help slow you down, instead of having to heat-brake everything.)

Suddenly, you're able to stock a ship, send it up, and say, "Hey, find the red planet in your telescope and just fly towards it manually." - Sure, it's not efficient, but it no longer needs to be. So instead of saying, "I need accurate time-clocks so my captains know where/when they and the planets are at all times", think of it as, "Ship captains know the relative position of the planets, and use telescopes to manually pilot their craft between them."

Realistically i can't think of a way to do this without a computer... a very complex mechanical system could in theory keep a match of all those different timezones on each different planet or moon, and then keep track of them based off a pendulum swing, but these systems will eventually go out of sync and lose time, keeping time accurately is a very difficult thing to do, that's why we have atomic clocks:

https://www.livescience.com/32660-how-does-an-atomic-clock-work.html

However

why were the computer's banned, presumably because they caused the downfall of Earth, but a multi-planet species is unlikely to ban computers, so there is only one real thing that could be used as a reason to band the survivors together and have them agree to ban computers... Religion

And while i don't want to start any form of religious debate, it is entirely plausible that the figureheads of this new religion could possess an Atomic Clock, now while this does rely on computers these days. they could either be Holy Artefacts or just that the religion is massively hypocritical. think of films such as Equilibium,

where emotions are banned, and that "religion" has clerics that go around and find and kill those that feel and destroy anything that could entcite emotion such as art. and yet the heads of state actually have lavish art decorating the walls etc.

You could easily expand they storyline in this way.