Was there radiation exposure to those handling the assembled bomb?
Probably some exposure, but probably not too much. Until you get near critical mass there is not high-level nuclear reaction. The Thin Man and Little Man used two slugs fired into each other and when the impact happened, the critical mass was exceeded and there was a nuclear explosion. It was the mass and geometry combined with neutron fission going to exponential levels that caused the explosion. The Fat Boy was a hollow spherical set of nuclear sections that were imploded into the center of the sphere where critical mass was exceeded. This was a difficult type of bomb to design because the triggers for the high explosive detonators to cause the implosion had to be timed so accurately.
No problem with assembled" bombs... However I wouldn't handle any of those archive pictures... It will be tough trying to pick your nose after your fingers begin to glow in the dark and then fall off...