I want to add a few things. Your timeline covers a lot of history. I myself have only read in depth about WW2 and the Soviet side of the space race.
* <PERSON> would make a little difference, but not a dramatic difference to Soviet rocket tech. Dramatic difference must come from dramatic more investment. (However, his absence from America could certainly cost them the Moon Race.) <PERSON> team built the F-1 engine because America invested in huge ground test equipment for such a huge engine, and in particular, fixed acoustic problems by literally detonating small explosives next to the engine pipes while the engine was firing to see if the shock would self-damp. If not, build a slightly different pipe shape. <PERSON> team was noted by NASA for its extreme meticulousness in assembling and inspecting the saturn rockets. The Saturn Family never exploded, from Saturn 1s to Saturn 5s. (But notice I am saying <PERSON> team instead of <PERSON> by himself.) See Challenge to Apollo by <PERSON> (caution: huge book).
* More USSR investment is plausible. They got all of Germany, so that's a lot of civilian labor and industry under their control.
* Just curious, but why did you put Soviet Moon Landing on 1970 Feb 14th? IRL, first man in space was 1961 Apr 12 but you said late 50's, so I'm assuming you're simply shifting the timeline forward. With <PERSON> and <PERSON> and more investment and <PERSON>'s same promise, don't see why Soviets would be delayed.
* The Outer Space Treaty...you should remove this from your timeline. TOST declares the Moon and other objects are not claimable, and it prevents nations from putting nukes into orbit. I think you want both of these enabled. Storing nukes in orbit would lower the enemies' reaction time by half. It could drive storing attack satellites in space too, and other countermeasures. It is therefore a great way to drive and keep space and rocket tech, much more so than commercial space mining activity or Mars mission. 199
Do you know how expensive space mining would be? Even rare-earth elements are cheaper to get on Earth. Helium-3 is not here, but I believe it can be synthesized in a nuclear reactor for much cheaper than going to the Moon. Besides, I believe Helium-3 can be collected directly from the solar wind, so only a simple orbital satellite is needed, not all the way to the Moon.
* If you wanna keep the manned Mars mission, do a Mars flyby or Mars orbital at the most. A Mars landing in 6 years is way too much, especially as America hasn't landed on Moon yet.
* As others have said, <PERSON> is limited to 2 terms, so put his brother <PERSON> next if you want. Would still take some explaining since <PERSON> just "wasted" huge resources on a failed Moon Race. IRL, the Space Race was actually rather unpopular in many fields like education because it diverted massive funds. Even the military did not like such a huge Saturn Rocket that was way, way too big for any nuke or spy sat.
* Just wanna point out that rocket tech and space tech will absolutely drive the computer industry, at least to the point that personal computers take off on their own. Spacecraft need the lightest, smallest, least power hungry comps available, because resources are so expensive in space. So no inconsistency there.
* Curious, you said space stations with refineries, but what are they refining? Only specific harvested thing you mentioned was helium-3, which needs no refining. Refineries on Earth---whether oil refineries, bauxite refineries, iron refineries, etc., are huge, huge, huge (heavy) facilities that need constant input/output of chemicals and usually go to molten temperatures.
* This may sound crazy, but a nuclear exchange does not necessarily lead to nuclear winter. IRL, America and USSR detonated hundreds of A-Bombs, even H-Bombs, above ground during their testing (roughly 1948-1963). (Really you should go by Megatonnes detonated per year rather than number of bombs per year, since MT/a went up hugely in the last 5 years or so IIRC.) This contaminated nearby areas, and some of them remain contaminated to this day, but did not threaten nuclear winter across the entire planet. See this table and Wikipedia's detailed list for more. But this can actually work in your favor and brings me to my next point...
* It seems pretty obvious to me that China was the one who stole/hacked the nuke and got US and SU to kill each other. That would greatly benefit China. Perhaps China planned to be the only one with the industry who can now save Earth with their rocket tech, and they want and will get total control of the new world. 435