Damn, can I manage to get it into "Fatherland" scenario. The threat of a nuke gets the Royals to flee but instead an invasion is the threat.
There are several ways to get to a "Fatherland" situation and they don't really need the removal of the Sovereign, who is after all a cypher as far as power is concerned. The importance of the Sovereign lays in the symbolic relationship with subjects (not citizens) of the Empire.
In “Fatherland” Britain was not occupied, but had been eliminated as a belligerent with a peace treaty that favoured Germany and installed a pro-German government--- these are optional ways to reach Robert Harris’s world.
1/. Weaken the agreements between Britain and France in 1938 making Britain more isolationist, slower to re-arm and less likely to make any commitments to Poland.
2/. Eliminate Churchill (obvious), send him to India in 1936 as Viceroy (he wouldn't accept anything less and would love to have made trouble there; he was sure that his Parliamentary career was over in 1935. Smuts was keen for him to be G-G of South Africa in lieu of a Royal, but a native South African was considered more politic), he is not in the Commons shouting warnings from 1936 to 1939 and cannot conspire against Chamberlain or be available to take over as PM or put backbone into the cabinet in May 1940.
3/. Have Britain sign a peace treaty after a failed Dunkirk evacuation on very favourable terms triggering a more rightist government that allies itself with Germany against “Bolshevism”.
4/. The USA, after milking Britain of all gold and other assets does not implement Lend-Lease and a flat-broke Britain has to declare a draw and signs a peace treaty in late 1941.
5/. As per Harris’s world, Germany wins the U-boat war any time before mid-1942 and a starved Britain has to sue for peace.
In order for Germany to knock-out the USSR it would be important for Britain to be eliminated from the war before 1942 and not supplying arms or intelligence to the Soviets and possibly actively aiding Germany. Most of all a USA looking the other way would be required.
If Britain went full-on fascist after a peace treaty, with opposition MPs being locked up indefinitely under Defence Regulation 18b, but Parliament still sitting and rubber-stamping, a 1960s Greek situation may have arisen with George VI running off to Canada as Constantine ran off to London, however the relationship of the Dominions with the Monarchy would be most problematic.