we would have world peace, happiness, and be conquering other galaxies
jestergooning

USA still would have somehow won, because that's what we do.
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At what stage were America developing the bomb, before or after declaring war?
. . . however, after considering this further, I did another google, and if the results are accurate, scientists (Lawrence Berkely to name one) were exploring nuclear fission in the early 1930s. So going by that, I guess one would have to say technically before declaring war.

I am not really good with history, but I did a quick google, and Wiki (not a source I would cite if I was writing some kind of academic paper, but for this date I think it will be okay) says that The Manhattan Project started in 1942, which was after the Japanese attack on Pearl Harbor in 1941.
As an aside, the "the backdoor to war theory" espouses the belief that FDR badly wanted to enter WW2 earlier but could not get support from congress or the population at large, so therefore he pushed a foreign policy which caused Japan to attack the US. The "backdoor theory" also suggests that the attack on Pearl Harbor was predicted by the government higher-ups, but the warning signs were purposely ignored, and Admiral Kimball, who was the commander of the Pacific Fleet at the time, was set up to be the scape goat.
That seems a bit farfetched to me; for one thing, I don't think that there were any guarantees that Hitler was going to honor the Tripartite Pact and back Japan's move, especially considering that believers in "the backdoor theory" also maintain that up until Pearl Harbor, Hitler had been going the extra mile NOT to provoke the US into entering the war.

At what stage were America developing the bomb, before or after declaring war?
Chopper: I'm just a bloody normal bloke. A normal bloke who likes a bit of torture

Pat Buchanon has made the argument that all America did by entering WW2 was to trade Hitler for Stalin.
As far as the outcome in the years that followed hypothetical, I would think that Hitler's invasion of Russia was one of the colossal mistakes that he made, and the best he could have hoped for after that was a stalemate.
As far as conquering America, he was unable to get across the English Channel or through mainland Russia, so it would appear that the United States was safe from a transatlantic invasion or him coming down through Alaska and Canada.
I have read varying accounts of how well his nuclear development program was going, and Hitler with a nuclear bomb would probably be nearly as dangerous to the world as the concept of trump with the so-called nuclear football is.

Hitler would have won and England would be a better place because he never would have allowed all of the migrants to come in and rape everyone.
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Russia would have still kicked the Nazis into the sun.
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And America wouldn't?
Chopper: I'm just a bloody normal bloke. A normal bloke who likes a bit of torture

You'd be speaking German today, Fraulein.
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