Complex topic. It would depend on the bomb in question. Some are more "dirty" than others.

The clouds of radioactive gases carry radiated dust particles that are carried by winds and settle on the ground, roofs, etc (fallout). That's why after Chernobyl or the Japanese cities were attacked it was very important which way the wind was pushing the clouds carrying the tiny debris, ash, and dust and how the Chernobyl disaster was detected by other countries in the path.

You probably also want to avoid trying to grow any crops in the area because one way to deal with the radioactive dust is to bury it under the top soil, and buildings that have been closed since Chernobyl that still have the dust trapped inside are still very dangerous.

Edit: the bombs in Japan exploded high above the ground to maximize damage and minimize fallout. The gases were carrying less radiated particles, and mostly dispersed after the initial blast or carried by winds. The gases over Chernobyl kept going until the fires were out.

It's been a couple of decades since I watched the documentary so maybe my memory is betraying me but from what I remember the bombs dropped on Japan didn't touch the ground. They detonated in the air so there technically isn't a ground zero.

Generally the really nasty gamma emitting fission products lose their nastiness after a couple of months. Their half lives tend to be counted in hours.