Nuclear propulsion in space pretty much relies on the fact that you're using
atomic bomb explosions to propel yourself around (Or in the case of NERVA engines, using stuff used to make atomic bombs). This could get a lot of people worried really fast.
To get this up and working again, you're going to need to find a way to end the Nuclear Test Ban Treaty and other possible treaties, and a workaround of the Outer Space Treaty. To do THAT, you're going to need to find some way to convince people it's okay to test nuclear explosives, above ground, where the explosions can have possible negative effects on health and definite effects on nearby electronics in the form of EMPs (or possibly just in space, still EMP-ing satellites), as long as it's for 'peaceful, non-weaponized purposes'. In the heat of the Cold War.
This is going to send the Soviet Union into full spook. The Americans have managed to test these massive, nuclear crafts, above ground and into space...and everybody's okay with this?! To avoid a nuclear panic and possible preemptive strike, the U.S would have to be reasonably open about their project, making sure to explain its method of propulsion in (reasonable) detail, and describing the payloads of such craft so as to mitigate suspicions of foul intent. The Russians are still not going to like this one bit, be it from the frightening nature of nuclear craft or the military supremacy of the U.S in space. Because of America's forced openness with their technology (and advertising how much more efficient it is than chemical rockets), the USSR, using combinations of observation, scientists and espionage, will try to piece together the fundamentals of the American nuclear drive and make their own. Eventually the Russians will begin testing of their own nuclear rockets, much to the dismay of the Americans.
The driving force of the space race and the Apollo program was the fear that Russia would gain military supremacy through space, and have a Russian military operation on the Moon ready to strike anywhere at any time. If Russia were to make it to the moon first, or show signs of continuing its space program, the U.S made plans to build a base on the Moon to counter a Russian base. After Neil Armstrong planted his boot into the lunar regolith, it became evident that Russia was beginning to lose its ability and drive to compete in space, and the plans of a moonbase were canceled.
A nuclear-enabled space race might get a whole lot fiercer, with Russia showing clear signs of overtaking the U.S. A Russian moon base might appear more clear than ever to the Americans, and plans of their own base might be constructed in more detail. If Russia sets foot first, the Americans will definitely try to follow suit, especially with their moonbase plans. If America sets foot first, Russia might still not lose hope with their new knowledge of nuclear propulsion opening up opportunities for vast amounts of resources to be transported long distances. They may not be the first people on the Moon, but an idea for a moonbase might be independently thought up by them (rather than being a simple fear of the Americans) and may attempt one. Optionally, they may wait until interest in U.S space exploration dies down (like it did in the 70's) and launch their moonbase mission without time for the US to react (Or perhaps they knew it all along through espionage and rolled out
their plan). Like I said before this would all be happening
in the heat of the Cold War.
If, somehow, you manage not to destroy modern civilization, there could be serious implications for space travel.
Certain designs of nuclear spacecraft are estimated to reach speeds of 0.3% to 3% the speed of light. And of course, there are the moonbases mentioned earlier. You know how much the Apollo landings ignited sci fi and dreams of young boys and girls to become astronauts? How the very dream of expanding technology couples with the advances learned from space propelled the OTL Space Age into cell phones and personal computers? Think of what's gonna happen with permanent or semi-permanent moonbases. Nothing ASB but it's gon be good. Manned missions to Mars suddenly seem more feasible (though you should still go through the same probe routine as in OTL). Probes can be conveniently sent throughout the solar system to investigate worlds. In a somewhat far future, a flyby of Alpha Centauri could be performed that could take less than a human lifetime in transit time. Scientific research would expound everywhere, and possibly a new appreciation of the stars and our place in the universe.
An increased likelihood of warfare in space is also possible. A destroyed satellite could cause a domino effect of creating even more debris, thus rendering humans incapable of safely leaving Earth for a good while. Simply blowing them up might be avoided in favor of creating craft designed to grab a satellite and deorbit it. The primary interests in space warfare might be orbital airstrikes (Likely not nuclear. We don't want to destroy civilization now do we?) and attacks on outer space outposts and bases.
And of course you're going to have lots of catastrophic accidents associated with nuclear spacecraft, just like how all the early spaceships had accidents (only these would be much worse depending on the circumstances). Doubt might arise about if nuclear power truly is the way to go for space. You'll have to find a way around this.
Nuclear power might be considered in greater detail for planetary use. These are early nuclear plants we're talking about though. An increase in plants may possibly mean more Chernobyls. More Chernobyls may mean an increased fear and dissidence of nuclear power, and such technology may be restricted to outer space use for the most part.
And, of course, there is the possibility of a hush-hush mission to get a fleet of ICBMs disguised as scientific (or commercial/logistical/???) spaceships looming over Earth poised to tactically and decisively knock out all rival nations' ability for defense and invade. This, I think, still allows the ability for a last-minute MAD.
All in all nuclear propulsion is a great way to get around in space and if it weren't for earthling squeamishness about putting potential nukes in orbit it would have put the space race on steroids. But also very dangerous as far as human fallibility goes.
I'm sort of new here, did I get too carried away? :\
