Would a strictly military program have lead to nuclear spacecraft?
I agree with the idea that making NASA the only US space program is a little backwards - it makes much more sense if it is considered the other way, with space activity in the US remaining strictly military.
In this case, there are some possibilities associated with Project Orion (spacecraft propelled by atomic bombs). Ulam first proposed it in 1947, and the project was started by Ted Taylor and Freeman Dyson at General Atomics in 1958. Now, OTL, Orion's funding was at first from the Air Force, which was interested in military applications. A few years afterwards, it was transferred to NASA and civilian ownership and terminated with the end of open-air bomb tests (I'd have to check the details in George Dyson's book on the history).
If space remained strictly military, and the US is competing in a space-weapons race with the USSR, then Orion spacecraft might have been launched in the early 1960's - putting more mass in orbit at once than everything before at least 1975 OTL. This would imply some re-writing of the Test Ban Treaty in 1963 and a somewhat higher rate of radioactive fallout. We may assume a bunch of interest in high-efficiency nukes, followed by lower fallout designs as the environmental effects become clear. A side effect of designing Orion pulse bombs is directional atomic field artillery.
If an Orion fleet or fleets (OTL, only the US was studying the idea, but things do leak) were to be built in ~1963, then a few missions to the Moon, Mars, and the asteroids around ~1970 are plausible. To give context: in 1958, General Atomic had drawn up plans for a crew of 20 to visit Enceladus (moon of Saturn), with a launch in the late 1960's and a return three years later. I like the image of a space-suited Dyson taking giant leaps across the geysers at Enceladus' south pole... Not sure I like the probable effects of six hundred missiles poised in rotating orbits over the USSR's population centers on mounts that use atomic detonations to run away from attackers.
Sorry if I've butterflied this analysis.