Say, for some reason, that WWII, or at least the Pacific War, lasts a few weeks or months longer, with the American President not wanting to use the bomb to make Japan surrender. Granted, the Japanese were too proud to surrender, but for this scenario, let's say that no bomb was used, and Japan eventually signed an ultimatum.
Let's just say that Japan is slightly less irrational, and surrenders without being nuked or invaded, somewhat later than OTL's V-J Day.
Now for the question: With the world not truly knowing the horrors that an atomic bomb can do, would it be more likely for nuclear bombs to be used in the Cold War, Cuban Missile Crisis or even a WWIII? Just a thought.
Well, first, the Cold War could be different if Japan's surrender is delayed. It's probable that the USSR occupies all of Korea, for instance.
Second, U.S. military strategy after the war was based largely on the idea that the U.S. had an irresistable air-delivered weapon, and didn't need any other military or naval forces. The next war would be a "push-button" war.
The Army, Navy, and Marine Corps were almost completely demobilized after the war, while the Air Force was elevated to a separate service.
In fact at the time, even more radical proposals were made - abolition of the Marine Corps and reduction of the Navy to a convoy escort force. There was a very bitter dispute over these questions. The Air Force tried to have naval aviation abolished (or merged into the Air Force, as had been the case in Britain between the wars).
If the Bomb has not been dropped - these policies would have far less impetus. The post-war drawdown would be less severe.
OTL, during the 1948 Berlin Crisis, U.S. policy was constrained by the fact that the U.S. Army had almost no fighting ability. There were at most five divisions that were fully manned and equipped. ATL, the U.S. will have retained a larger conventional-arms force, and may be more willing to risk open war.
By OTL 1950 the U.S. had recognized the folly of trying to rely exclusively on air power, but the U.S. was still seriously under prepared for Korea. ATL U.S. can put up a better fight.
As to the use of the Bomb: it would be used somewhere as a tactical weapon.