No way a German "overlord" in 1940
You may be right on the politics, but Germany wasn't nearly going to be ready for an amphibious assault on Britain in 1940. While their ability to improvise with inadequate logistics resources and inadequate logistics planning was admirable, that was pretty much at the breaking point when they stopped (the point about not grabbing Dunkirk being a good one though; that was within reach). Even another major campaign on land would have been out at that point. Take a look at what d-Day required and compare what the Germans could have had available in June of 1940. No go, not if you assume a Britain that has resolved to resist (which is required -- otherwise, why invade if you can get a surrender?).