A race between America and the USSR of making nuclear engines would fit in well with one of the main reasons the space race happened in the first place, which was to get the better missiles to nuke the enemy quicker in a nuclear war
Nuclear "missile" engines were dropped pretty early in the planning because the needed thrust-to-weight wasn't there due to the nuclear engine mass. (Even without shielding) The initial idea of using a nuclear engine for an ICBM was from the military misunderstanding (which was a general problem with the public and a LOT of writers who should have known better
) of how a nuclear reactor worked. The militaries general idea was to use the 'warhead' as a reactor till it got the target then have it blow up, this was pointed out by the AEC and most military physicists to NOT be the way either a reactor or a bomb actually worked. Couple that with the low T/W (despite the theoretical high ISP) the military soon dropped the idea.
Also on the whole Nerva thing since it shares a name with a Roman Emperor maybe generations of nuclear rocket engines are named after Roman Emperors (Nerva 1st Gen, Trajan 2nd Gen, ect.)
NERVA (which was an acronym) was given by the AEC not the military standing for "Nuclear Engine for Rocket Vehicle Applications" and was meant to study a wide range of nuclear powered rocket designs. The engines themselves were given different designations and names depending on the proposed applications or experiments. (A 'small' engine named "PeeWee" or one designed to explode being named "TNT"
)
In fact there WAS competition between nuclear engine design teams with the NERVA 'beating" the likely more capable "
Dumbo" design (scroll down to "Dumbo" entry) mostly due to NIH (the former coming from the more weapons design oriented Los Almost nuclear engineers) and a disbelief that nuclear weapons designers could also design a rocket engine. (And keep in mind another set of those "weapons designers" were playing around with "Orion" at the time as well
)
Randy