He wouldn't have been out reviewing the bomb damage or telling people to keep a stiff upper lip and give it back, that's for sure.

I think he would have continued to try to work behind the scenes to make a peace deal. After the Battle of Britain, when it was obvious that an invasion of Great Britain wasn't in the cards, he might have become more openly a proponent of "Peace with Honor". This would have left Hitler free from any worries about the Western Front, North Africa or Italy and able to send even more troops east.

And a British King with such obvious pro-Hitler views would have energized the American isolationist movement. Also if Hitler had someone like Edward around he might have prevailed on the Japanese not to launch a sneak attack and risk bringing America into the war.