Easy: When Baldwin tells Edward that one of them had to go, Edward tells Baldwin: "You're it, I'm keeping her and the throne", since Baldwin had said he'd resign if Edward tried hanging on. Attlee had gotten caucus backing on the subject, he wouldn't form a government either. So there would have to be a snap election which would result in another Tory landslide. They would then pass an Abdication Bill declaring the throne vacant with all the precedent using the nuclear option would create. After all, everyone remembers the last nuclear option- Asquith's.
The problem with these scenarios is that Edward was a spineless coward and wanted Simpson much more than the throne. Baldwin knew this, which is why I don't know if he even ever discussed said abdication bill despite being the logical post-election outcome. Edward can hang on as king but not with Simpson around. Everything else aside, on this issue Baldwin had a spine, Chamberlain even mroe so (he wanted to turn the screws harder and faster). Edward doesn't. As soon as they corner him and turn the screws he'll back down.