Bombs can be "tuned" to burn nearly all or very little of their fissable material. Radiation from the blast dissipiates nearly immediately, but the bomb would need to be designed to provide little fallout(it's doable is what I'm trying to say).
Amount of fallout depends on more things, not just settings of the bomb. Mainly on where the bomb detonates. If close to ground/water, then it's bad, lots of fallout. If in stratosphere, very little fallout.
If you burn less, you lower the yield. Burning more means you turn more mildly-radioactive plutonium or U235 into intensely radioactive fission products. Strontium-90, for example, accumulates in bones and has a half-life of 28 years.
