Not necessarily, obviously. As nuclear explosions go, the blasts at Hiroshima and Nagasaki were pretty clean. In fact, in the weeks following the bombings, American authorities trying to keep a lid on the deteriorating PR situation portrayed A-bomb damage as being just like that from conventional weapons, except that there was more of it. Those already dying of "atomic sickness" knew better. Only gradually did the world realize that, even if you can safely walk through the ruins of a bombed city soon afterward, the effects of a nuclear attack continue to show up for years.

Atom bombs like the ones dropped on Japan produce two types of radiation: initial and residual. (I'm getting this from Hiroshima and Nagasaki: The Physical, Medical and Social Effects of the Atomic Bombings, an exhaustive Japanese study, published in English in 1981.) Initial radiation is released by the explosion itself. Residual radiation comes later from radionuclides, radioactive isotopes either generated by the explosion or else induced in soil, building materials, bodies, etc, by neutron bombardment unleashed by the blast.

The bombs that destroyed Hiroshima and Nagasaki produced their share of residual radiation, but it didn't stick around long, for two reasons. First, both bombs were detonated more than 500 meters above street level so as to wreak maximum destruction (surrounding buildings would have blocked much of the force of ground-level explosions). That limited surface contamination, since most of the radioactive debris was carried off in the mushroom cloud instead of being embedded in the earth. There was plenty of lethal fallout in the form of "ashes of death" and "black rain," but it was spread over a fairly wide area.