The Cold War was based on nuclear armament, and once the Soviets had the technology to make an atomic they were determined to do so. We (US) were also building up a nuclear arsenal to counter what the USSR might do. Hard to say, but no one would back down. Of course, we think of the USSR as the aggressor, and they think the opposite!
We DID explode the first bomb, but both the US and USSR were close to development, and bombing Japan was not the primary struggle for the US, dealing with the USSR after the War was.
So probably no, unfortunately. It escalated during the Cuban Missile Crisis, but there was little trust between the two superpowers since even before WW II.
