For the scenario to actually work, i'm going to make some basic assumptions.
1. All state heads give their complete and unilateral support for the project.
2. Nazi Germany might exist earlier - otherwise a ww2 situation wouldn't really work, alongside some real life brain drain mechanics not working.
3. Theres essentially a hard limit of 1936-1937 anyway because the understanding of nuclear fission wasn't advanced enough before then.
To be entirely honest, the basic points have been stated on this thread already.
Nazi Germany probably isn't getting any nukes. Driving out basically all the scientists essential for a atom bomb program basically killed the Nazi bomb project before it had really begun. However, one of the killing blows was due to high command assuming it would be a short war. Remove that, and you have more potential. It should be noted that in this situation, its assumed that all great powers of the time would work on a atomic bomb project - And that would include France. Maybe French scientists and material captured during the fall of France would help timewise. No idea.
The US has a more debatable outcome. If they were too start by themselves in 1936 or 1937 and persist by themselves without allied help, They might be able to have a functioning nuclear weapon by late 1944, early 1945. On the other hand, OTL the Manhattan project was only really possible with the knowledge addition of the British "tube alloys" project, alongside various Canadian scientists, and of course, all of the scientists fleeing from Germany and Italy. Assuming allied cooperation, mid or early 1944 might be possible. Early 1945 would definitely be possible.
I don't really see the UK making an atomic bomb by themselves. OTL the tube alloys project was merged with the Americans for a reason - Britain just didn't have the industrial infrastructure to create conventional and nuclear weaponry at the same time, so that job was passed to the US. With this in mind, if the UK were to continue to produce an atom bomb themselves, I can't really see them making one before late 1945, but probably not at all. With allied cooperation, it would basically turn into an American project anyway.
French couldn't really complete anything before they get invaded by Germany. As previously discussed, any scientists, equipment, and scientific theories would probably go to the German atomic program.
The Japanese probably couldn't get anything functioning before the end of the war. OTL the Japanese atomic program was held down mostly by rivalries and a poor industrial capacity, not unlike Germany. The army and the navy had separate nuclear programs, both of which ended up at the same place at the end of the war. Scientist wise Japan wasn't much better than Germany, but because Japan wasn't against using Jewish or other expelled scientists, and while the did officially have a policy of antisemitism, in practice it wasn't actively enforced, and anyone who helped the Japanese war cause was overlooked. Given all of those factors, a bomb might be possible late 1945, but that honestly still seems unlikely.
The Soviet atom bomb project could be a little more successful due to Atomic spies, but I still can't see anything happening ww2. Maybe an earlier first test, most likely early 1947, possibly late 1946. Aside from that, not much could really be accomplished. Soviet nuclear industry was essentially non-existent, and the fact that they built an atomic bomb 1948 OTL was already a miracle, as American and British intelligence didn't expect a soviet nuke until early 1950s.
TL;DR - It doesn't really change. The situations I put forward were already stretching it as they were. OTL there was a reason that only America made nukes: they were basically the only country that actually could. The German project was doomed from the start, The Japanese never put anything towards a serious effort, The British project didn't have the industrial base, and the only nuclear industry the Soviets had was their spies. In this situation however, the Americans might get more nukes earlier, which would be interesting to see how that played out.