Mutations are caused by errors in DNA repair mechanisms. Radioactive particles smash into DNA, breaking the bonds that hold it together. You have a whole system for DNA repair in your cells, but it's not 100% accurate, and so the more times you have to repair things, the more errors you're going to see.
And the reason you'll never see it in random frozen samples from 20 years ago is that your cells have to be alive for it to happen, and those ones aren't.