Would the British surrender if London were captured or move to a second city? Manchester, Birmingham..? Would Churchill resign? Oswald Mosely puppet PM?
Think of it of a sort of a reverse Valkyrie. Himmler, with a massive paramilitary organisation, takes power. He even coerced Goering to relinquish control of the Gestapo. So he'd have no problem with Goebbels.
Operation Sealion succeeds in 1940. The Einsatzgruppen are released into Britain. Who are to be arrested? I.E. Winston Churchill?
Sources: McNab (2009). Other sources would be extremely appreciated.