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.
That should be the Little Boy and the Fat Man bomb names.
fascinating bump
With the success of the Trinity test some of the scientists wanted to strip the thin man and make it into eight fat boys. The fat boy trigger was necessary for the plutonium bomb because the time constants associated with plutonium fusion meant it would tear itself apart before most of the fuel fissioned in a thin man configuration. The fat boy was a more efficient trigger but the thin man was such a simple design they felt their was no need to test it.
The military felt the urgency of delivering a blow was more important than having more weapons later. Their calculations were correct. Two bombs sufficed.