if I remember correctly, the Nazis had trouble picking out potential collaborators among the British population. Try as the could, they couldn't find anyone who they thought would be a perfect puppet for them.
That being said, if push comes to shove, I think they would've asked King Edward to come back on the throne. Edward hated being Governor of the Bahamas. Also during a visit to Germany in October 1937, he was treated with the upmost respect by the Nazi hierarchy and was believed by the former Austrian ambassador to the League Nations, Albert von Mensdorff, that Edward favoured German fascism as a bulwark against communism, and even that he initially favoured an alliance with Germany.
As for Oswald Mosely, I see him being something of a puppet Prime Minister, but I believe the Nazis would put all of the power into their puppet king and turn Britain into something of a faux absolute monarchy, with Mosely just being used as a conduit between the King and the British people. Before the war, Mosely, though he was a member of the British upper class, held no serious political pedigree and took more inspiration from Mussolini than Hitler, so I don't see why he would be the Nazis' top man in London.
I think Free France would cease all hostilities with the Axis if London had fallen. The British were the only thing supporting De Gaulle's exiled army. If the British had surrendered, I don't see De Gaulle being able to put up a fight. He would either flee to Canada or maybe, get captured by the Nazis and deported to Vichy France, where he would be executed for treason. But the flame of French resistance would've been snuffed out if Britain had fallen. This goes for the Polish and Czech undergrounds too. With no support from the UK, they would've fallen apart due to lack of supplies.
Ireland was a neutral nation, so I think the Nazis would more or less leave it alone. Maybe they give Northern Ireland back to shore up support among the Irish population. Maybe they depose the Irish government and install someone from the IRA? Who knows?
As for the colonies, (by which I assume you include the Dominions), I don't see them waving the white flag and recognizing some false king in London. They would recognize King George as the true monarch, ruling in exile from Canada or maybe the Bahamas, and continue the fight.
As for Scotland and Wales, I think the Nazis would keep them bound with England. it would be much easier to occupy one, united and controlled nation than three, disorderly nations.