Circuitry and radioactivity don’t mix. Bots might be the solution, nanos probably not.

Nanos would face the same issue living things face - ionizing radiation taking out vital components in one shot.

What if they were made of lead? Or do you need a thick layer of lead to shield from ionizing radiation?

Nanobots by definition have important/active parts with sizes on the scale of molecules. Doesn't matter what the molecules are exactly - if a radiation particle hits an atom and ionizes it, it's likely to break the molecule it's in and thus the mechanism. By break, I don't mean break apart - but a molecule that suddenly gains or loses charge in its structure starts interacting differently with other molecules.

(Except it's how ionizing radiation damages DNA; cells implement error detection, error correction and self-destruction mechanisms to limit the spread of DNA damage. Unless we go the biotech route and make our first nanotech by repurposing viruses and bacteria, our nanobots are unlikely to be as resilient.)