I don't see it, the worst that'd happen to Edward VIII if he tried to hang on to the crown and the woman, is Parliament "Impeaching" him, which surely would be a crisis for the monarchy and leave it much weakened, and maybe break up the Commonwealth, truthfully if Edward VIII hangs on, keeping Willis on the side and being a dickard I could see the Monarchy being very much weakened, PMs basically ignoring the King, the week meeting becoming 5 minutes of tea, relations between the other monarchies and the sovereign break down maybe ending the Monarchy in the 1960s or at Edward's death
but 1940s not happening, the Monarchy is too deeply rooted and supported in the 1930s and 40s, it'd take decades of it being undermined and mismanaged to get from that support to overwhelming Republicanism, and it would need to be overwhelming, in Australia Republicanism has had up to 60% support and of course has never happened