Assess the impact of nuclear weapons on state rivalries and relations after World War II. How did the proliferation of these weapons affect Soviet-American relations?

In many respects, nuclear weapons raised the stakes of military and political conflict, due to the overwhelming destructive power they represented. From the moment that the technology was unleashed, via the success of the Manhattan Project, the global calculus of politics was forever changed.
In the aftermath of World War II, the United States was in the unique position of being the world's only nuclear power. Militarily, relative to competing states, one might say this is the most powerful that the United States has ever been, but it also resulted in a kind of diplomatic and political nightmare situation, because the very weapons that had allowed the United States to achieve such power within the world were also weapons that could potentially be one day used against it. Meanwhile, by the very nature of the technology, all other competing powers (especially the Soviet Union) could not, as a matter of national security, allow the United States to maintain that nuclear monopoly. The imbalance in military power was too great.
This resulted in the arms race, with the United States maintaining a short-lived nuclear monopoly and the Soviets racing to catch up. The Soviets would successfully test their own atomic bomb in 1949, while Britain would follow in 1952. Of course, this was not the end to the arms race, for the atomic bomb was really just the first foray into the technology, and so the technology would continue to advance. The result was the arms race between the United States and the Soviet Union. Neither nation could afford to fall behind.
I would suggest this was a critical and defining feature of the Cold War. It resulted in a very tense contest, but by the very nature of the technology involved, it also discouraged direct military conflict between the two sides. Ultimately, a direct military encounter would invite escalation toward a nuclear exchange, which would leave both sides devastated. Indeed, this threat of escalation did rear its head at times (most famously with the Cuban Missile Crisis), but regardless, it was not by accident that most of the military conflicts between the United States and the Soviet Union were fought by way of proxies. Direct confrontation represented an existential threat to both sides.