There's a lot to consider. I'm not sure I'd make the same assumptions you have.
First, where we I agree:
I can accept that events in the Pacific will develop pretty much as OTL. Japan can't win and I see nothing in your scenario to suggest the Nazi-Soviet alliance would necessarily have reason to help them any more than in OTL...although it might get interesting if Japan becomes part of what amounts to be an "Eastasian/Eurasian" alliance against "Oceania".
I could also see the allies accepting something far less than unconditional surrender from Japan ... maybe even accepting an armistice similar to what was offered Italy in OTL.
I also expect that a war like this would have a number of effects on both the USA and UK that would make them more conservative than in OTL (economically, socially, and in terms of race/gender equality). Britain would consider preservation of the Empire essential, and putting the USSR in what amounts to be the enemy camp, would completely discredit strongly left-leaning movements in both the US and UK. This would adversely effect all causes seen as revolutionary of leftist, such as racial equality, economic levelling, expanded women's rights - unless they were taken up by non-leftist organizations such as corporations or the church. This is possible, I suppose. By "republican triumph" are you talking about the US political party or disestablishment of the royal familiy in Britain? In the first case, my answer is yes, in the second most definitately no.
You seem to be imagining a war around the periphery of the Nazi-Soviet landmass (the middle east, North Africa, Scandinavia, maybe even the over the remnants of the former Japanese Empire) rather than an allied attempt to invade Europe massively and hastily. This makes sense to me.
Now where we disagree:
First, If the Nazi and Soviet non-aggression pact survives they would form an economic block at least as strong as the UK/US alliance, especially when one considers that, even in the 1940's the Empire was more of a drain on Britain than an asset. I tend to think that the UK/US would see this block as a much more significant threat to their survival than either Germany or the USSR alone. I am not convinced the allies would feel secure in the faith they could outproduce them.
Second, I believe this would hasten, not retard the development and deployment of atomic weapons by the US/UK alliance. While you may be right that allied control of the air over western Europe might be less certain, I don't see why that would inhibit the use of nuclear weapons - they'd just be used differently - possibly in areas where local air superiority coud be temporarily obtained. First, I'm not sure nuclear weapons would be used on a prostrate Japan at all in this TL, when the real enemy is still out there and very powerful, so the very existence of these weapons would be unkown to the enemy. Because of the risks of having bombers trying to reach Berlin with atomic weapons could lead to their capture, I suggest that the allies might chose to use these weapons, not as demonstration city killers, but as the immediate prelude a high-stakes military operation such as an invasion of western Europe. The "shock and awe" provided by the detonation of a dozen or so 35 kiloton bombs among German defenses and rear areas in Normandy would not only decimate the defences but might have unanticipated effects on the loyalty of Nazi satellites such as France and Italy, plus make the Russians begin to rethink the wisdom of their alliance with Germany. Since the dangers of radiation and fallout would be only poorly appreciated, US and British forces would pour ashore against an already defeated defending force. By the time the allies realized that they had underestimated risks to their own people from radiation exposure, the front line German forces would already be defeated and fresh alied forces better equipped and trained to avoid "hot" areas would be coming ashore. Lots of poshumous medals for the initial wave though.
Probably a more likely outcome of the geopolitical world you describe, however, is just a different sort of Cold War. The Angloshphere Alliance may decide that the cost to defeat a Nazi/Soviet alliance is just too much, offer an armistice to Hitler, hope that they two dictators will eventually fall out and fight to the death, and then pick the time and place to finish them off.