What makes you think von Neumann got any exposure in labs? He was a mathematician, not any sort of experimentalist.
The idea that his cancer came from fallout is just moralizing nonsense, the silly logical error of thinking that the universe operates in a way that punishes badness.
He was actually at Los Alamos, he was trained as a chemical engineer; it’s hard to imagine he didn’t get invited to see the demon core and other fun stuff, or otherwise breathe in some particles.
So, you think being in the room with the "demon core" was dangerous even when it wasn't being used? The radiation from it in a non-critical state would have been very small. If he had been there during a criticality accident it would have been recorded.
I think you're just waving your hands here because you have no evidence to support the claim.
Everyone (and many family members) working at SSFL 15 years later died early of contamination. They were careless in those days. Probably had a few buckets of the stuff laying around like at the Grand Canyon gift shop.
The idea that his cancer came from fallout is just moralizing nonsense, the silly logical error of thinking that the universe operates in a way that punishes badness.