Where is the proof that exposure to radiation increases cancer risk throughout life?

Idiots claim that nuclear bombs and reactor meltdowns lead to massive cancer outbreaks. But where's the proof? For Fukushima there's only 1 reported death, from lung cancer, which is very questionable and no evidence that it was caused by radiation. For Chernobyl, there is proof only for 28 people who died, all within weeks after extreme exposure from radiation sickness.

Yet there are claims that people are still dying as consequence of Hiroshima and Nagasaki. Where's the proof? How does it make sense that exposure to radiation will cause cancer only 50 years later?