The short answer is, sort of! Having a monopoly on nukes was a major reassurance to American foreign policy while it lasted. Soviet successful testing of an atomic bomb was a key moment that changed US calculus in its grand strategy, and it greatly disturbed the Truman administration, further indicating how important nuclear monopoly was for the time. But did the US use that nuclear monopoly to make major moves and gain huge concessions from enemies and allies? That is less clear.

The thing to note here is that at the close of WW2, the US was by far the winner and predominant power. Sure, the Soviet Union was a military power too. But most foreign policy analysts projected that the US would have about 5-10 years of great flexibility and leverage before the USSR recovered from all its destruction during the war. Economically, diplomatically, militarily, the US was extraordinarily more powerful than its main rival. Even without nuclear weapons, most American officials would have felt that they were on top because, materially, the US was in far better shape than any other comparable power.

The atomic bomb, while certainly a game changer, was relatively primitive at the time. It had to be dropped from a B-29, and it was only really good for demolishing cities. That worked well with military doctrine during WW2, total war could justify targeting civilian centers and economic hubs. The notion of a tactical nuke, one that could assist the battlefield in less dramatic ways, was not technologically available yet. Furthermore, the US nuclear arsenal was relatively small at this point, so threatening to wipe a nation off the face of the earth was not possible yet.

So what did nuclear monopoly do? The effect was mostly psychological. American officials could feel confident that they could win any major war that erupted at the time. Dramatic moves like the Marshall Plan and NATO, which would certainly provoke the Soviet Union, were okay risks because the Truman administration could count on the Soviets recognizing the supremacy of US power in all categories, nuclear and otherwise.

Why did the US simply threaten to destroy Moscow in order to get its way with regard to Eastern Europe? The thing to note here is that the Cold War was more of an idea than a geopolitical reality in the early years after WW2. For much of late 1945 and 1946, officials in the Truman administration were torn between aggressively pushing against Soviet interests and continuing cooperation when possible in the spirit of the wartime alliance. There was a growing rivalry, certainly. Even when a consensus began to emerge, the favored strategy was to contain the Soviet Union and communism, and allow their system to collapse of its own accord. Many scholars have argued that it is really only with the Korean War in 1950 that most people start to see the situation as a "Cold War."

The loss of nuclear monopoly, however, was very startling. Soviet possession of the bomb came sooner than expected and shook up US projections of the timeline for when complete supremacy of US power would expire. Poking the Soviet bear got much more complicated and risky. Pressure mounted in the Truman administration to accelerate work on a bigger, better bomb (the H-bomb), one that could correspondingly cow Moscow into recognizing American technological and warfighting superiority. More and better delivery systems were developed that could send nukes into the interior of an enemy country from a safe distance. Thus, an arms race began. It is only really until the 1960s that the dangers of relentless nuclear proliferation start to be mutually understood, as both nations have "second strike" capabilities.

To summarize, nuclear monopoly was important for the US. Faith in US supremacy and predominant power was part of the confidence that led the US to take risky actions in Europe and elsewhere. But the conditions of the early Cold War meant that there were fewer war hawks who called for the complete and utter destruction of the "enemy". The limitations of the US nuclear arsenal also meant that nukes had few uses. It took some time for a consensus to emerge that the US needed to confront the Soviets, and even then, containment was the strategy of choice.

Sources: Melvyn Leffler has an extremely detailed tome on every moment of the Truman administration in A Preponderance of Power. The introduction of Masuda Hajimu's first book, Cold War Crucible, has a great discussion of when the "Cold War" becomes "capitalized", as 'The Cold War' as opposed to simply a cold war.

Aside from the more threatening aspects of the nuclear monopoly, there was also moves towards so-called "international control" of atomic energy. Part of Cold War's onset this was to do with who would have control of this terrifying new technology. Within the Truman administration, there were those such as Undersecretary of State Dean Acheson and adviser David Lilienthal who in late 1945 and into 1946 pushed for international control. In essence, this meant that production and storage of atomic technology would be handed over to the new United Nations, who would control it for the good of all humanity.

Needless to say, there were others in the US and UK who saw the atom bomb as the lynchpin of future military power and resisted attempts to internationalise the bomb. The so-called Acheson-Lilienthal Plan for international control metamorphosed into something quite different. Truman appointed as his lead negotiator in this the Wall Street financier and rabid anti-communist Bernard Baruch. Under Baruch, the plan changed significantly during the first half of 1946, becoming known in popular parlance as the Baruch Plan.

Although the plan that was put in front of the United Nations Atomic Energy Commission (UNAEC) in June 1946 proposed international control, it required the Soviets to hand over any and all atomic technology before the US would give up its bombs AND mandated that the United Nations Security Council (UNSC) members would not be able to veto any sanctions related to atomic weapons. It also mandated intrusive inspections and the ability for the international community to undertake military action against those who attempted to develop the bomb. Needless to say, the Soviets were not at all keen on this, proposing their own counterplan, the Gromyko Plan (which was - rather interestingly - modelled on the 1925 Geneva Protocol for the Prohibition of the Use in War of Asphyxiating, Poisonous or other Gases, and of Bacteriological Methods of Warfare), which was rejected outright by the US. Ten out of twelve members of the UNAEC voted in favour of the Baruch Plan BUT Poland and the USSR abstained. As the measure required a unanimous vote to pass, it therefore went nowhere. So, the dream of international control died in the face of the Truman administration’s burgeoning anti-communism, the personal whims of Bernard Baruch, and Soviet reluctance to submit to the plan.

Hope this helps.

Malcolm

Reading

Campbell Craig and Sergey Radchenko, The Atomic Bomb and the Origins of the Cold War (New Haven: Yale University Press, 2008)

Ronald E. Powaski, March to Armageddon: the United States and the Nuclear Arms Race, 1939 to the present (New York: Oxford University Press, 1987)

Shane J. Maddock, Nuclear Apartheid: The Quest for American Atomic Supremacy from World War II to the Present (Chapel Hill: University of North Carolina Press, 2010)