In some sense, the real answer here is probably unknowable. Two events happened at nearly the same time, and they were closely followed by a third. Figuring out conclusively what caused what may simply not be possible.
Something not often drawn into this discussion is the long-term impact of Hiroshima and Nagasaki. In August 1945, there was still some sense, even in the Japanese high command, that the destruction of Hiroshima was not much different in kind that the destruction of Dresden or Tokyo by conventional bombing. It took, maybe, gradual understanding of and empathy for the actual horrors of nuclear warfare, and the experience of the Japanese cities and their survivors, to create a global reluctance to actually use nuclear weapons again -- a reluctance that has lasted 75 years so far.
Assume no atom bombing and a quick Japanese surrender ending the war. Without the image of Hiroshima and Nagasaki, how much greater the likelihood that some Cold War Strangelove would have gone ahead with use of tactical nuclear weaponry, or the hydrogen bomb, or the Doomsday Machine?
Is it acceptable to think that the destruction of those cities spared the rest of us from unknowable horrors? Deep waters.