The crews that flew the planes that dropped the atomic bombs were not regular bombing crews. They were members of the 509th Composite Group, which had trained for a year or so in the art of dropping very large, awkwardly shaped bombs ("Pumpkins," which were atomic bomb casings filled with only high explosives, essentially) from specially modified B-29s ("Silverplate") that had their armaments and armor stripped from them. They also told their work was totally secret compared to other bombing groups, and they accompanied other bombing raids with their "Pumpkins" in the run-up to the late summer of 1945, and trained in flying essentially solo raids (one or two B-29s at most, as opposed to hundreds) of this sort as well.

So they were not random at all, and they knew they were up to something. Some of them later reported that they had figured some of it out. The captain of the Enola Gay, Paul Tibbets, of course knew the whole story as he had been involved in the Manhattan Project on several ballistics and delivery questions.

So that gives a fair account of what they needed to know: they needed to know how to deliver large bombs accurately to targets from high altitudes, unaccompanied by other bombers or fighters, in defenseless bombers. They needed to know that their work was important despite its appearance of being unimportant, and that they needed to keep it quiet.

Michael Gordin's Five Days in August has a lot about the 509th in it; John Coster-Mullen's self-published Atom Bombs has a whole lot (but is harder to track down).

Side question (I also heard the interview on NPR yesterday) the pilot mentions taking a few pictures from his plane as soon as he could unstrap from his seat. Do those pictures exist anywhere?

Just a quick correction for OP, NPRs guest was a crew member on Necessary Evil, not Enola Gay

Did any of the crew of either nuclear bombings express regret later in their lives?