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.
Not consenting to a police search doesn’t stop the search, and that’s ok.
What it does is make the fruits of that search inadmissable, and may also enable you to sue them if the search was unreasonable or excessive, or the pretext violated your rights.
Even if you know you don’t have anything in your car, verbally and clearly say that you don’t consent to the search, and would like them to note that fact, but otherwise comply. Lots of people have been caught up by police planting evidence, and you don’t want to be one of them.