The biggest problem with the scenario is the United States and British Empire were reaching the limits of their resources, and the population was exhausted from the years of sacrifices and casualties.
The British actually considered something similar with Operation Unthinkable, but concluded that even in the strained conditions of 1945, the USSR would still could not be defeated, and that the British public would never support a continuation of the war on these terms. The Americans were coming to similar conclusions when considering different plans for the defeat of the Japanese Empire - the US Navy's proposal to simply surround the Home Islands and starve them into surrender by 1948 was rejected out of hand, and there was considerable doubt that Operation Downfall, the proposed invasion of the Home Islands would be acceptable to the American public with the projected casualty count of 250,000.
Nuclear weapons were certainly the way to break the impasse (which is why Harry S Truman authorized their use against the Japanese Empire), but even the United States, with all her resources was strained by the vast expense of both the Manhattan Project and the development of the B-29 bomber. Without the B-29, the delivery of nuclear weapons (outside of exploding them in the harbour aboard a ship).
The real issue here is that the height of America's "Imperial" impulses actually occurred prior to the Great War, with America gathering an overseas empire after the Spanish American War and securing interests in China during the Boxer rebellion. An aggressive President Theodor Roosevelt might be capable of rousing America to continue attempting the conquest of the world, but in this case, a rising America will meet the European Empires at the hight of their powers (before they were consumed in the Great War).
On the other hand, the United States may actually have accomplished your aim in the post 1945 environment without a shot. The American dollar became the world reserve currency, the Americans created and nurtured the institutions like the Bretton Woods Agreement, the IMF, the WTO and even the United Nations to bind the world into a global order of her own making.
So perhaps you need to reframe your story. Conquest doesn't necessarily need to be at the point of the sword, and America can arguably be said to have conquered the world (or a large fraction of it) between 1945 and the 1970's, and have still has enourmous influence to this day. There were a few occasions where the Americans might have actually used nuclear weapons in battle (the Korean war, the 1962 Cuban Missile Crisis and perhaps in the 1973 Yom Kippur war), but you can see in the real word this wasn't really necessary.
My advice might be to take one of these potential triggers and explore the consequences of actually using nuclear weapons then.