The only radiation sunscreen protects you against is near UV light not radioactive emission
Sunscreens are designed to protect from the near UV radiation present in sunlight. Some block both UVA (315-400nm) and UVB (280-315nm) light but some don't work well for UVA. All UV light at ground level is harmful to skin to some extent and too much of it will cause skin damage that looks a little like a burn (though small amounts are good for encouraging vitamin D production).
Radioactivity can cause similar looking burns. But the cause of the burns is not near UV light but gamma rays (hard X-rays of much shorter wavelengths and much higher energy than UV light) and alpha and beta particles. All of these are much more dangerous than UV light (as the amount of energy deposited in tissues is far greater per photon or particle).
And sunscreen will offer no protection at all to radioactive emissions. Clothing can protect against alpha particles. But you will need several millimetres of aluminium to stop beta particles (and even then the secondary gamma rays from the interaction of those particles with the aluminium can be harmful). And gamma rays might need a thick chunk of lead to keep them out.