I recall reading about the possibility of nuclear-powered command & control planes also. I don’t know if that idea ever made it to anything you could even call a planning stage.
Its possible. But they would have to land anyhow when they run out food and water.Alternative underground command posts were built in Pennsylvania, Colorado, Virginia (two), and West Virginia.
When a nuclear powered aircraft crashes — and they will crash at sometime trough mechanical failure or human mistakes — the chances of the reactor core being exposed are very great. How many countries want exposed reactor cores on their property? Zero. Even the Russians never accepted the idea of a split open reactor spewing radioactive contamination around. (And that's saying something for Russian's desire for survival.)