I recognize the fact that the US had nukes and felt that would save them some time but why didn't they just want to finish the job early rather than over the next several decades with the Cold War?
An annihilatory attack on the USSR was a seriously discussed in many parts of American society at the time. Some of the proponents were surprising - Bertrand Russell [1] for instance. There was also a major movement to put atomic weapons under control of the United Nations e.g [2]. I can't find any overall discussion of contemporary thought in the UK (which would have had a veto at this time), but it appears that Churchill thought such a war necessary [3].
A practical problem was that it turned out that the USA did not have any bombs. This was discovered by Lilienthal, first head of the AEC, when he visited Los Alamos in 1947 [4]. "Actually, we had one [bomb] that was probably operable when I first went off to Los Alamos: one that had a good chance of being operable". Apparently most of the the technicians had left after the war, many diagrams and plans had been lost (including all details of the gun-type uranium bomb), and the plutonium bomb had never been "weaponised" to make it possible to assemble it quickly, store it for more than a few days, and transport it without a serious risk of unwanted explosion.
References below are taken from "Command and Control" by Eric Schlosser. This is a popularised account and when I checked some of his references, they didn't match what he put in the text - hence I've noted what I've checked myself and removed some references that didn't tally.
[1] "Russell Urges West to Fight Russia Now", New York Times, Nov 21, 1948 (I have only read the abstract due to the paywall).
[2] "The Atomic Bomb and the Prevention of War", Bulletin of the Atomic Scientists, Oct 1, 1946 (checked)
[3] Trachtenberg, "History and Strategy" (not read)
[4] Herken, "Winning Weapon" (checked)
There was no thought of the United States allying with Nazi Germany against Russia.
When Germany's defeat was imminent (before the A bomb) Churchill apparently ordered a study of whether a British -US alliance could defeat the Soviet Union called 'operation unthinkable'. The study concluded that it would be a long war and against the odds, and generally a bad idea.
The Soviets had a huge advantage on the ground with more armored divisions, more artillery, more everything. Their soldiers and leaders were experienced, competent, and willing to endure horrible losses and conditions. Their equipment was also mostly good by 1945. The Soviets had a tactical airforce instead of a strategic force, but they had more fighters in the west than the western allies.
Without going too far into 'what if' territory, maybe a small number of A bombs makes a difference, maybe someone resorts to newly discovered nerve gas... I think under any scenario, this would have been a worse outcome than the subsequent Cold War struggle.
Thought/relevant question.
Can someone tell me the size of the Allied air forces compared to the Soviet ones in 1945?
I assume that the Soviets could have held air supremacy over at least the Soviet Union against the Allies. Nuclear bombs at the time were delivered by bombers. The US would have to get bombers past what was perhaps the most powerful air force in the world at the time.
Could nukes have been realistically brought to bear against the Soviet Union at that time?