Overall, what can we say about whether there would have been more or less war in, say, the 1950s and 1960s had WW2 been won with only conventional means, and the technology for nuclear weapons never manifested for whatever reason? Did the A-bomb stave off another industrialized world war, or did the resulting American hegemony and Cold War simply funnel the conflicts into proxy wars that might not otherwise have occurred?
This is a very hard question to answer.
From a data point of view, the world's wars, as measured as per capita deaths from military activity, has decreased a lot since World War II. Now World War II was a bloodbath so that's not too hard to have accomplished. But in general per capita deaths from military action are pretty low since WWII even compared to previous centuries.
It's not really clear that per capita is the best way to think about this, given that the population of the Earth has gone up by a factor of about four since World War II. Wars aren't spread equally, so booming populations will automatically drop the average number.
And if you dig into those numbers you find that the number of small wars has increased since World War II, including the number of revolutions and other kinds of instabilities.
Separately, if you do want to say that since 1945 there has been less war, do you attribute it to the atomic bomb? While it is clear that the bomb had some effects on world order, it isn't clear how much of that is attributable to it and not other factors. Why not, say, attribute that change to the United Nations, which were formed at the same time as the bomb? Or some other aspect of the postwar world, like NATO? How would you distinguish such things from the effect of the bomb? Should you even try?
How should one separate out the potential deaths of hypothetical conventional World Wars, whose existence is not at all guaranteed "without" the bomb, from the real deaths of the Korean, Vietnam, and Afghan conflicts, to name just a three of the big proxy wars?
I don't know the answer, and frankly I distrust anyone who makes a strong assertion that the bomb or anything else is responsible for such a thing. We live in a dynamic, multi-causal world, and single-cause answers are suspicious in any case. And I also think it takes a pretty careful manipulation of data to come to the conclusion that we live in a world of less turmoil and violence as the past — the amount of violence and wars is still pretty high, and just because it doesn't touch as large a percentage of lives doesn't mean it's worth patting ourselves on the back about.