As I'm working on a fictional setting based on Turtledove's "The Road Not Taken", I've spent a fair amount of time of late trying to tackle the question of accomplishing things in space without a proper spacefaring technological base.
I bet you've come up with really interesting workarounds to achieve results with low tech! Either your own ideas or ones you've picked up from others working this quite interesting problem. I'd love to hear about them.
I can't help but wonder, based on the technical discussions had thus far, if the thread isn't a bit hung-up on the aesthetics of the Apollo program.
If by "the thread" you mean "me," you have a point. (There isn't much of a coherent thread happing here yet, just little cliques talking past each other. I point no finger of guilt at anyone without three of my fingers pointing back at me!

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The reason for my Apollo "obsession" being that Apollo was, OTL, the minimum bare-bones "get 'er done" approach NASA adopted.
Obviously anyone looking at Apollo can point to swathes of "fat" to cut to make it leaner and meaner--after the fact, we could probably do it a bit closer to the edge, with what we know now (not talking about the general advance of technology since designs were mostly frozen in the mid-60s, talking about what we know about the actual challenges the space environment presented and the actual logistics of operating on the Moon--"do it again" in that if we ISOTed some notable experts with some kind of ASB aphasia about actual technology but reliable knowledge about space, Apollo could have been cheaper and faster in the 1960s).
But what some might call fat, I'd call safety margin. Paring it down a lot means an even riskier mission.
In terms of the basic mission profile, I don't think you can pare the essential steps involved in getting x number of human beings onto the Lunar surface and then back home again alive to Earth down more efficiently than the Apollo program arrived at. The conceptual alternatives all boil down to even more mass launched into orbit, at whatever state of the art we are talking about. Disposable stages, Lunar orbit rendezvous, direct return to Earth from lunar orbit--these are the essential features of Apollo, and when the Soviets considered routes to the Moon themselves they either chose a more massive brute-force direct descent and return model (Chelomei) or a mission that breaks down essentially to a pared-down Apollo (Korolev's N-1 launched Soyuz-LK mission).
So it's in that sense that I'm sticking to an Apollo script, because the alternatives are even more costly.
For instance, do you need functional radio to reach the Moon? Viscerally it seems obvious that such is required, as you can't have Mission Control without it, and you need a Mission Control to do anything in space. But taking a step back, is there anything genuinely requires radio? Not really: The only thing I can think of is the radio altimeter for the lunar descent, where it emoves the guesswork from trying to land the descent craft.
Exactly the point I was stuck on! I agree, we can have a knocking good romantic story of steely-minded, steady-handed space adventurers who are also top-notch Newtonian physicists do it all in silence, hopefully living to tell the tale when they get home again. Except for that pesky approach to the Lunar surface!
Can it be done by eyeball? That's not a rhetorical question I'm asking; it's a puzzle. The thing is, the human eye is great for operating on Earth, even doing something as outlandish as flying (and most to the point here, landing) an airplane when no one has ever done it before. But we are used to seeing things in an environment where the atmosphere literally colors everything, diffracting light into shadowed zones, lending gradients of hue and diffusive blurring that give us vital cues as to how far away things are.
If only Earth had a second moon, a much smaller one on the scale of say Phobos, a captured asteroid or something like that. They could do an earlier mission to there, and "landing" would almost be more a matter of "pulling up to;" the point is they'd have lots of time to practice, with trial and error, learning to correctly read visual cues in a vacuum. And when there is confusion of the visual cue the horizon provides, fooling us into thinking we are farther from the surface than we are because the horizon of a smaller body is closer.
Remember, the point of getting a moon landing done is that someone has to do it
first. That someone gets no benefit of advice from a seasoned, practice veteran who has done it before and learned by trial and error. The first pilot has to get it right the first time, or they don't live to coach rookies.
And that brings me to another aspect of electronics:
Without it, we have no automated probes. If we did have some sort of clockwork automation but no radio to report findings to ground control, we might as well have nothing. Well, almost.
Consider what the preliminary scouting of the Lunar environment did for the success of Apollo OTL. Until the Soviets and Americans started shooting robots at the Moon, there were some grave unanswered questions shadowing the prospect of a human landing. For instance, what could be the nature of the Lunar surface? We might guess it is a lot of bare rock--but that's not quite the case. In fact the surface of the Moon is some kind of dust, fines--but fortunately they stick together pretty well by various adhesive forces. Until we sent down lander probes, it occurred to some people that the Moon dust might be very slippery, being in vacuum with no atmospheric gases or moisture to mediate chemical bonds. The bottoms of craters might be pools of dust behaving like oil--and any spacecraft that touches down, despite being landed with uncanny skill and smoothness, might promptly sink right into it and vanish beneath the smooth surface!

Well, we know now that air or no air, the gravel, dust and fines do stick to each other after all, giving a powdery surface more like brown sugar than oil. That sure is good to know, isn't it? Without some kind of radio and the potential for automated probes that implies, and automated probes signaling back their findings, only human-piloted vessels can land, and only a surviving human can carry back what they learn when they try. (A wrecked lander might have survivors who can manage to put on spacesuits and go out and set up a heliotrope or something to signal their own epitaphs by Morse code. I don't know that they could see any replies, unless from another spaceship in close Lunar orbit).
What I want to learn is that yes, it can be done by human eyeball, and of course the first lander will be flown by people who know that unknown unknowns, as well as known unknowns, might well kill them--they are called adventurers for a reason.
But I still think it's daunting and we should give respectful consideration to hitches such as this.
Lack of communication can be an issue, but ship-to-ship -- or even ship-to-ground, with a big enough telescope -- communication can be accomplished just as was with wet navies at the time, signal hoists and light-based signalling. (Zero-G semaphore is the cooler of them, methinks.) It's not terribly pretty, but it worked well enough for the Royal Navy until the advent of wireless telegraphy.
Especially because if we don't have an Apollo analog mission, the most likely way to succeed is with a very gradually building space program--first they put up some satellites (crewed of course); then build a proper space station or more; then eventually, over decades, build up the infrastructure in LEO to make the longer jump to Lunar space--then arrive in orbit, stay in orbit, make an orbiting Lunar staging base, and finally attempt landing with not one Command Module astronaut watching them, but a network of several orbiting stations all with several astronauts in them, watching the descent through telescopes and for semaphore or heliotrope signals. Or of course the minimal version is broadly similar to Apollo and so there is anyway the one astronaut left in the command/return craft peering anxiously through his telescope at his companions.
What radio is primarily used for is controlling and directing a flight: Removing it simply changes the character of astronauts. Astronauts would, by their nature, have to be creative, fast-thinking, and comfortable flying by the seat of their pants, as every problem after they hit orbit they'd have to deal with on their own. The Royal Navy offers an interesting parallel from prior to the advent of radio, in the cult of the god-captain, who presided over distant patrols in the South Seas. One would expects that something comparable might evolve to facilitate dealing with the time in orbit, as it would not be dissimilar in effect from being on a ship halfway between Australia and New Zealand.
Interesting discussions can also be had about the strict necessity of advanced electronics (i.e. can bravery and a slide-rule be a good-enough substitute for a flight computer?), but I think it is ultimately moot.
Yep, I agree, they can get along without those new-fangled computational engine doohickys. Radar would still be handy though. Gonna have to make do with a steely gaze I guess.
You just need too much materials science that just isn't there in a Victorian tech base to build a rocket capable of putting the payloads necessary into orbit to accomplish a Lunar landing.
Actually I don't think they need any materials that couldn't be made in the late 19th century, really. It's just that doing things with what they had then would mass more. And it adds up, so since we could hardly get by with 1960s tech at less than say 2/3 the all-up mass of a fully loaded Saturn V, with all the incremental add-ons, with reserve fuel making up for roughly estimated calculations, with heavier materials substituting for lighter, with less efficient rockets requiring more propellant to do the same job and a need to do things in less efficient ways to allow for more margin of error and slower response times...it adds up, then the lower ISPs mean it multiplies and exponentiates--so we wind up needing a monster rocket 4 times more massive than a Saturn V, or dozens of smaller rockets to laboriously build up the moon ship in orbit (using less efficient but more storable fuels instead of the more efficient ones that won't keep for months or years while we assemble the thing), or running the whole Moon mission on the back of a really massive orbital infrastructure....one way or another the project is absolutely more costly than Apollo was, by a factor of ten or so, and in a world that is not nearly as rich altogether as the OTL USA of the 1960s was.
And yeah we might identify a materials Achilles heel yet, I suppose. I still haven't thought of a good way to make pumps to drive a liquid fuel rocket, or even the liquid part of a hybrid, that wouldn't also mean these "Victorians" also have jet airplanes. Pressure feeding might still work but it involves a lot of overhead mass, for the tankage (not as much as you'd think though) and the pressurant (that's still a bit confused in my head.) I still like the idea of boiling liquid hydrogen to get the pressurant, but then we need something like pretty good plastic bags for the hydrogen peroxide oxidant, so...work in progress, and lately I haven't been thinking about it as much, having chased down all the easier leads and being left with some serious engineering questions.
But I remain optimistic on the materials front. It's just a question of, what motivates and enables this colossal expense, on such a risky project?
Where's the payoff?
You can nudge a PoD here or there to get some of it, but the only way to get all of it is to: 1) create a timeline that has a civilization with a post-1950s tech base by 1899 located in the modern British Isles but which is otherwise alien to ours; or 2) introduce a form of phlebtonium that removes the issue of reaching LEO but all chucks out the Standard Model as we know it.
Actually I'm trying to suss out how to do it with as little tech wanking as possible, so the late 19th century society doing it (IIRC by the way the OP never said it had to be Britain...oh wait, yes they did...in the title actually...

...oh well, if any single nation in a world resembling OTL 1890s can do it, it is Britain anyway, though by then the Germans and possibly sufficiently motivated Americans would be in the running too...OP certainly never said neither or even both of them might help)---anyway, a world with as little advanced over OTL and yet still make it possible. Not cheap, but anyway possible.
That's why I want to avoid turbopumps, you see. And electronics, if we can do without it. That's why despite the modest ISP, peroxide-oxidized hybrid paraffin or "candy" rockets may be the way to go.
Still haven't let go of the insanity of peroxide burning hydrogen though.

I'm pretty sure we need pumps for that; the question is, are they tantamount to having jet engine parts, or can it be done sort of Gothic?
The lack of radar rangefinders is a problem, but not insurmountable.
OK, I said it was an open question--how exactly do you foresee surmounting it, then? Some very clever sighting instrument that takes the guesswork out of visual measurement of rate of descent, or just plain seat of the pants eyeballing it?
The latter can work, I guess, if 1) the pilot is mentally prepared to remember the vacuum of the Lunar surface will look different and the Moon's smaller radius hence closer horizons need to be borne in mind and 2) a good amount of reserve fuel. That's one of the places where we trade off having resources to waste (because we splurged relative to Apollo's mass budget, by paying a huge price in launched mass) for lack of more efficient methods. If we've precalculated correctly, a big burn can bring the lander to a near halt a safe margin above the Lunar surface, then it starts to fall again, but we've got a hopefully throttleable engine of some sort--the pilot turns it up when it looks like they are coming down too fast, and down when it looks like they are taking too long to get down, balancing the whole thing on trained intuition.
Hopefully they can manage some sort of clockpunk simulator for the pilot to play Moon Lander with a lot until they get the rhythm down.

I agree with Juumanistra, we do not need radio.
I hope. As others say, having radio of some sort, at least enough for a crude altimeter (probably based more on frequency shifting than direct measurement of pulse return times, some sort of interferometer) would actually not be that difficult to get, but then the world is less Victorian and more Jazz Age, even 30s-40s vibe.
Now, how about this. Maybe we don't need a heat shield. What if we put a rocket in LEO with the 13,000 Delta-V necessary for a direct landing? When the astronauts come home they can dock with the reentry rocket and bring it down.
Oh dear Lord. Sort of doable, but it is a classic instance of what I was talking about, making the net mission mass exponentially greater.
Look, a big driver of the whole mission mass is the Translunar Injection--that means basically taking something in low Earth orbit and boosting it so it is going just a hair under escape velocity. Doing that once with Apollo's J-2 hydrogen-oxygen engine, a more efficient engine than anything we can hope for in 1900, more than doubled the total mass that had to be orbited. With lower ISP engines, that doubling can easily become a tripling (50 percent all up mission mass rise) or quadrupling (doubling the launch load). This is a price we have to pay to get anything to the Moon at all, there is no way around it and the higher masses in the retro scenario are also inevitable.
In order for explorers returning from the vicinity of the Moon to rendezvous with anything parked in low Earth orbit, they have to do it again. That is the velocity they have to lose, coming down from the Moon, is the same as that they had to gain to go there. And it's a big price item. So if Apollo's J-2 engine allowed them to keep the mass multiplier down to say 2.2 for that stage, their electing to instead return the Command Module to an orbiting space station instead of just letting aerobrake in Earth's atmosphere would mean that the net multiplier goes up to 4.8, to 5 or so. We'd have needed 2.5 Saturn V launches to do it!
That's bad. Trying to rocket-land from orbital speed to avoid the risks of aerobraking is even worse; it involves needing propellant to achieve a velocity change quite as large as everything above taken together and more.
Meanwhile--achieving reentry the way all spacecraft of OTL have done, by entering the atmosphere and dumping the orbital (even translunar) energy as the form of heat does not require any ultramodern, high-tech materials. We used the expensive high-tech material approach on the Space Shuttle because we didn't want any of it to erode away; it had to be reusable as is, you see. But allowing ablation, the erosion of material by heating it up enough to vaporize and then letting that material blow away in the slipstream, taking the heat away with it, means we can use a much wider range of materials.
Water for instance. Suppose we make the bottom of our capsule a water tank, of strong high-temperature steel, but with the very bottom perforated with little vent holes, plugged by material that melts and softens at high temperatures. Now as the capsule enters the atmosphere, the water is warmed, then starts to boil--this sets the upper limit on the temperature the steel handles; we have steam escaping the tank thus cooling it to maintain temperature; indeed the escaping steam forms a layer between the steel and the hottest incoming plasma.
Or--cork. Cork has been proposed as a serious candidate for heat shield material.
The point is, if it is inferior to say the fiberglass-acrylic laminate used on Apollo, that just means we need more of it by mass, thus raising the launch mass. But not by the ungodly amount we'd need to do it on rocket thrust!
