Depends how late it happens. If Edward had sudden attack of sanity, and decided to not marry Simpson (preferably also dump her, after its proven to him that she cheated on him), then the ever stiff-upper-lipped British society might be willing to pretend crisis never happened, glad it's over and they won't have to deal with mess of abdication. But I assume OP refers to Edward hanging onto crown despite sticking with Simpson.
Now, the closest thing to Edward winning here, is deciding to settle on common-law-marriage with her. As far as cabinet would care, it'd be legally equivalent to her remaining royal mistress, so they could live with it.