For lead the radioactivity comes from uranium mixed with the original ore, not from outside sources. That's why it needs to be hundreds or thousands of years old rather than just pre 1945.

>All lead mined on Earth naturally contains some amount of the radioactive element uranium 235, which decays, over time, into another radioactive element, a version of lead called lead 210. When lead ore is first processed, it is purified and most of the uranium is removed. Whatever lead 210 is already present begins to break down, with half of it decaying on average every 22 years. In Roman lead almost all of the lead 210 has already decayed, whereas in lead mined today, it is just beginning to decay.

https://www.scientificamerican.com/article/ancient-roman-lea...