The History Channel did a what if a couple years ago called Nazi Britain. Where they played with the idear that Operation Sealion the occupation of Britain had been undertaken.
It speculated that the Nazis would have occupied the Southern part of England including Wales, and that the remains of British Army would have hold on in Northern England and Scotland. The program also speculated that the Germans would have inserted Edward as king in London, and that the British royal family would have taken refuge in Canada. I wonder if this had taken place how would the e.g. a Liberatin of England by the US have taken place?