Nuclear power in a space rocket? There is no way that would be allowed: as you may know one of the main problems with nuclear power (fission to be exact) is waste products, and where to put them. There was the rather obvious idea of ejecting them to space. How? With a rocket. Why was this not taken up? The great dangers of a rocket malfunctioning just after launch, exploding, and the resulting radiation spread over a large area of the world...
And nuclear power only provides electricity (actually heat to be more precise, which is then usually converted). Ever seen an electric powered rocket? (Ok, it is possible to do a space shuttle on Jumbo Jet style launch, i.e. as an aircraft turning to space vehicle, but the chances of that being approved are also low, as well as the practical problem of needing much more time to get out the atmosphere.) The main problem is that there is no propulsion method which can use electricity for space travel. You can use propellers in fluids such as air/water (i.e. what 'planes and ships do), but in vacuum that's useless.
The NERVA (Nuclear thermal rocket) method, which uses a nuclear reactor to heat a propellant for propulsion is also not that amazing, and when NERVA was abandoned, it was still less effective than chemical engines. Maybe it could be improved, but it would hardly bring much more than a chemical engine; it's theoretical limit isn't amazingly greater than chemical engines.
The reason the space race is over is since we no longer have the cold war, which was the main driving force for it all. Space travel is expensive, uneconomical, and more importantly, useless. There is little benefit to be had from space travel for humanity.