The only case of anyone militarily involved in the bombing of Hiroshima and Nagasaki having mental health issues was Claude Eatherly, who was the pilot of the Straight Flush, the weather plane involved in the Hiroshima mission (they confirmed the target had good weather over it, and radioed back to the Enola Gay, who then bombed it). He was arrested in the postwar for various petty crimes, was for a time put into a mental institution, and invoked his role in the Hiroshima mission as a justification for his guilt and problems. He was extremely unusual in this; almost all other members of the 509th Composite Group affirmed their approval of the atomic bombings until their deaths, and had developed strong rationales (ending the war, saving lives, etc.) for their actions.

For more details on Eatherly, see Ronnie Dugger, Dark Star: Hiroshima Reconsidered in the Life of Claude Eatherly (London: Victor Gollancz, 1967). For a very probing discussion of the bombings of Hiroshima and Nagasaki and their psychological dimensions more broadly, see Robert J. Lifton and Gregg Mitchell, Hiroshima in America: Fifty Years of Denial (New York: Putnam's Sons, 1995). I don't agree with every conclusion they make in the book but they do a very good job of cataloging psychological impacts on some of the participants, and on others people. But the military people directly involved in the mission were mostly without regrets (and one can understand that — they saw their role as important, they did not shy from the deaths that come with military operations of that sort).