Depends heavily on the kind (and intensity) of radiation. Beta (electron/positron) and gamma (photon) generally won’t, but neutron and alpha can. Many of the atoms that become radioactive will rapidly decay, and that’s one of the mechanisms behind the impact to structural integrity.

Love the cheese name! I think he looks like more like a Muenster to me, but Colby works! https://en.m.wikipedia.org/wiki/Muenster_cheese