There are huge variations in what’s been proposed.
A nuclear-powered airplane would most likely use a closed-cycle reactor with the heat replacing the combustion in otherwise fairly conventional jet engines. They’d be totally harmless in normal operation, with the radiological danger being if they crashed and scattered the contents of their reactor.
Similarly, nuclear thermal rockets like NERVA are closed cycle and pose no radiological danger unless they explode or crash.
And then there are open-cycle designs such as nuclear ramjets, fission-fragment rockets, and Orion. Those are bad news bears for anyone nearby, and possibly the entire planet.
I mean even the "worst" of all designs, the one linked in the comment we're all replying to(SLAM - ramjet design) says this:
" Specifically, he states "The reactor radiations, while intense, do not lead to problems with personnel who happen to be under such a power plant passing overhead at flight speed even for very low altitudes." In both documents, he describes calculations that prove the safety of the reactor and its negligible release of fission products compared to the background. Along the same vein of these calculations, the missile would be moving too quickly to expose any living things to prolonged radiation needed to induce radiation sickness. This is due to the relatively low population of neutrons that would make it to the ground per kilometer, for a vehicle traveling at several hundred meters per second. Any radioactive fuel elements within the reactor itself would be contained and not stripped by the air to reach the ground"
Orion is obviously incredibly bad due to the fact it uses actual nuclear detonations for prepulsion. But it's never been a very serious project, while SLAM has been built and tested.
Of the things I listed, SLAM seems like the least bad. The reactor at least tries to keep the fission products within the reactor. That said, I would be very skeptical of safety claims from someone who needs it to be safe for their project to be successful, especially from the 1950s.
It's worth remembering that airplanes have to start, land and taxi as well. So while radiation levels might be safe while the plane is flying at altitude, things might be very different where planes have to land and start.
A nuclear-powered airplane would most likely use a closed-cycle reactor with the heat replacing the combustion in otherwise fairly conventional jet engines. They’d be totally harmless in normal operation, with the radiological danger being if they crashed and scattered the contents of their reactor.
Similarly, nuclear thermal rockets like NERVA are closed cycle and pose no radiological danger unless they explode or crash.
And then there are open-cycle designs such as nuclear ramjets, fission-fragment rockets, and Orion. Those are bad news bears for anyone nearby, and possibly the entire planet.