Anyway ignoring pro-Nazi revisionism, my take on the question.
In the event of the British being smashed at Dunkirk and lacking the power to oppose Hitler directly, signing a peace treaty might have been fairly likely. Electing a far-right government is not. Going to war with the US is even more insane for both the UK and Germany.
The German have defeated the British and about to defeat the French, they are free to fight the war that they want to fight, the Eastern front. Why would they want to start a war against the Americans? In the OTL the Germans declared war on the US partly because the US was helping the UK. They wouldn't have that reason.
The UK would have what remains of its army free to deal with the Japanese. With no UK-US alliance, the Japanese might not need to attack Pearl Harbor and bring the US into the war.
Even if the combined British and German navies could defeat the US navy, what would they do next? They could hardly invade the US. Neither the British nor the Germans would have enough land forces. The logic of the British deciding that they can't oppose Hitler because they don't have the land forces, but believing themselves to have enough to conquer the US is one that escapes me.