I'd imagine the manned space programme is abandoned in the 60s. By the 70s there will be a resurgence of interest in setting up weather, communications and spy satellites - although everything will be ten years behind OTL.
In the 80s I guess it's possible that there will be some sort of Low Earth Orbit military spaceplane programme - like the programmes which were abandoned in the 60s and 70s in OTL when the shuttle programme replaced them.
Interesting and a special area of interest of mine. At the time, the last manned spaceflight before the war (by the US) would have been Wally Schirra's MA-8 flight of "Sigma 7". (the final Mercury/Atlas orbital flight, Gordo Cooper's "Faith 7" occurred on May 15, 1963 in our timeline... sadly the Mercury program would have ended seven month's earlier with the outbreak of the war-- the federal government and the country in general would have had MUCH more important things to worry about than space flight, and would continue to for a LONG time to come...
The final spaceflights of the old Soviet Union occurred over a five-day period from August 11 to 15, 1962... the dual flights of Vostoks 3 and 4. Cosmonaut Andrian Nikolaev completes 64 orbits and Pavel Popovich, launched a day later, completes 48. The two craft come within 5 kilometers of each other despite no orbital manuevering capability, which Soviet propaganda used to tout they had "rendezvoused" in the first dual-spacecraft manned flight, which was actually accomplished (orbital rendezvous) by the Americans on the flights of Gemini 6 and 7 on December 15/16, 1965. With the destruction of the industrial infrastructure, research, academia, and general society of the Soviet state in the war, it would likely be a century or more before any Russian would again travel into space, on a Russian vehicle anyway...
While work WAS already being done that would lay the foundations of the Gemini follow-on program that would follow the Mercury program, and lay the path technologically to achieving the Apollo lunar missions, and in fact some early work was already underway on defining and designing the Apollo spacecraft and missions, and many development projects (like Saturn I and the F-1 engines) were well underway, the simple fact is that all that would have been halted in short order by a general nuclear war in October of 1962. Without a Soviet enemy to demonstrate technological mastery over to the world, the moon goal so eloquently set out by President Kennedy in May 25, 1961 would have fallen to the wayside in light of more pressing concerns like Reconstruction and re-emerging from the war-induced Super Depression...
However, in considering TTL, I can see how a damaged yet recovering America, humbled and yet re-emerging from the difficulties, austerity, malaise, unrest, and national humiliation following the War, and sentimental and pining for the "good old days" pre-War when America could "do anything", and looking for a way to assert itself on the world stage in a non-military way, would embrace the idea of a "bold new adventure" in returning AMERICANS to space... Perhaps pick up the mantle of technologies and materials developed for Gemini and Apollo before the war, reuse and adapt what still existed, and move forward with the goal to put an AMERICAN back into space by 1977 aboard a US capsule. I don't think it's tenable to consider this program being contemplated much before 1977, 15 years after the war-- any sooner and the problems facing the country would have made such an endeavor seem particularly wasteful and expensive to a still suffering American population still mired in the lingering effects of the Super Depression and war recovery and facing the challenges of civil and governmental unrest, which would make such a program look like a fool's errand considering the more pressing problems "here on Earth".
However, some 15 years after the War, with the US economy austere and stuck in malaise but revived (much like the late 30's I'd imagine, functional but weak, before the industrial revival induced by war trade and war work shattered the lingering effects of the Depression once and for all), I could see such a program being proposed by a forward-thinking and visionary President as part of some sort of "New Deal" type program of government projects designed to stimulate science, industry, and the economy all at the same time... sort of like a latter-day WPA (which surely would have been re-instituted in the aftermath of the War to salvage the economy and get people back to work through some sort of Reconstruction Act...) By now the economy has revived, but this will invigorate it, give the nation a goal and something to have pride in, re-assert American political power and technical acumen to the world which it has been largely sidelined from for a decade or more, and yet be non-threatening and foster goodwill at the same time, being a scientific program.
Simply re-establishing the American capability to orbit a spacecraft would have of course only been a preliminary step. A goal would be needed, and landing a man on the moon could be seen as a realistic and challenging objective, if it could be divorced from the stigma of it's origins in the pre-War Soviet/American Space Race... (which would depend on the political acumen and skills of the President proposing such a program after the War, and his popularity and charisma with the electorate). Perhaps a smaller step, establishing a permanent US Space Station, something more realistic and easier to achieve technologically and fiscally, and not linked to the Pre-War Space Race, might be a more 'palatable' objective politically-- so much would depend on the cultural attitudes and predilections and fears of the general population after the War, which is impossible to predict or accurately speculate with any certainty... If the population saw the Space Race as a "fools errand" that ENHANCED that competition (and indirectly led to the attitudes that caused the War) then I'd say proposing a Moon Program would be politically very risky. If, however, the population instead nursed a spirit of melancholy and a strong desire to "relive the glory-days of America before the War", to "return America to "it's rightful place", then I could see the announcement of a "renewed Lunar Plan" as generating a lot of excitement and pride...
It would almost surely be a capsule-based approach though-- the work on military spaceplanes like Dyna-Soar weren't especially far along in 1962, when the War would have cut into such work quite visciously and curtailed it for the immediate indeterminate future... Capsules like Mercury (and Vostock for that matter) were proven to work, and would be less technologically risky. As a follow-on program for the US after a "new Gemini" to return the US to manned spaceflight capability and achieve the goals and skills needed for further spaceflight (remember that in TTL no man had EVER walked in space (Leonov's spacewalk on Voskhod 2 didn't occur until March 18/19, 1965, and White's spacewalk on Gemini 4 on June 3, 1965 both occurred well after the War had wreaked havoc on the world). After the US returns to space in the early 80's and gets it's "Space legs" back, what vehicle follows on after that largely depends on the Goal... A space plane like Dyna-Soar for manned transport to a space station would probably be quite feasible, but if the goal is lunar exploration, then a follow-on capsule would be a MUCH better choice and is basically required to go beyond Earth orbit, so something much more like Apollo than Dyna-Soar or the space shuttle would be required...
As to whether China, India, or Japan would be motivated to engage in some sort of "space race" in TTL really is open to debate... I'd imagine ALL of these nations would have pursued some kind of launch vehicle/satellite programs... spy satellites would prove VITAL for national defense and intelligence gathering, and the communications capabilities for both the military and civilian uses would prove very hard to ignore. Also, burgeoning communications capabilities offered by satellites would have proven quite lucrative, especially for countries looking to replace damaged or destroyed surface-based communications infrastructures (cable telephone systems) or create new infrastructure that hasn't existed before. Such would have been a powerful motivator to pursue launch vehicle and orbital satellite capabilities in MANY countries capable of it, including the US, China, India, Japan, Australia, and equally motivating in less space-capable countries like Israel, Brazil, Argentina, and South Africa. England would be too emaciated financially and technologically to attempt it, though they might well have their needs served by Australia or India, and of course ESA would never exist as we know it. France might pursue satellite launch capability, depending on how badly they were hurt. Without doubt, the US would have maintained SOME capability in unmanned space launches for national security concerns, both for spy satellites like "Corona" of that era (which were launched on Thor/Agena rockets) and other military/intelligence assets (like Samos) and military communications satellites, electronic intelligence satellites, and other such assets which were all being developed or early models were already in use at the time, and have proved both disproportionately highly effective and valuable and also relatively invulnerable. In the immediate aftermath of the war, though, MANNED spaceflight would SURELY have been seen as a luxury and distraction that simply could NOT be afforded.
I've thought about writing something just along these lines...
Later! OL JR