This. Fallout is the radiation given off from the bomb when it explodes, it poisons the water, the animals the tree's, and more importantly, you.
The Survivors from Nagasaki were very sick after the bomb due to Fallout and many died, which is why im suprised there isn't any deformities with charecter's in Fallout 3 because it effects people in Japan today.
Yees but in Fallout people don't die of lukemia and cancer due to radiation exposure. They either die of 'radiation poisoning' (which in Fallout seems to be a threshold of Rad intake), or they ghoulify (after a slower, longer method of exposure). In Fallout, considering the scenario, the most devastation would have been suffered during the great cleansing, if rural areas weren't hit that hard, or at all, they would have at least had the time to take an initiative before the fallout started touching down from the atmosphere, and then it would have been a matter of years (or even months) before most of the lethal rads would have died out. The same generation would have at least been able to try and rebuild after the great war, much like the Hub just 16 years after the great war, and most control Vaults were scheduled to open roughly 10 years after the great war.