Only way him not abdicate is that he never marry Wallis Simpson. Marriage with divorcee was only reason why Edward VIII was enforced to abdicate. If that doesn't happen it wouldn't change much. British monarch hadn't anymore real power so there is not much what he couldn't do. Even his Nazi sympathies wouldn't matter. Government just bar him to speak to BBC or other medias.
Lets take a step back here, this isn't a game of A Very British Civil War after all.
The first question is, why does he change his mind and dig in? However once it becomes clear that the King will not abdicate, its pretty clear whats going to happen.
The first action will be that the government will resign and call an election explicitly over the issue. Baldwin was a pretty savvy operator and he's going to win a massive majority. Baldwin will also draw into the idea of maintaining the concept of a National Government (even if its overwhelming Conservative) and Labour nor the Liberals are going to support the King - Attlee's and Sinclair's OTL actions and responses would suggest they wouldn't. This is going to be 1918's Coupon Election in overdrive with the big three parties all aligned to send a message to the King that parliament authority is supreme, and that the will of the nation is against him.
So on balance, the parties are not going to hemorrhage many supporters or members to a mooted King's party. This is trouble with any notion of a Kings Party is that it’s inherently unstable and on any other issue/policy than the King, so even if enough MPs could align on the issue, it’s more than likely to collapse and fall apart, given the wide spectrum of members it would have. It would lack any coherent policies or objectives. It lacks the local political infrastructure of the established party machines.
Well that understanding is incorrect. Most reliable assessments of Edward's political views show nothing outside of what was generally "acceptable" within the British social, economic and political establishment during the 1930s in terms of British views towards Anglo-German relations and attitude towards the peoples of the British Empire.
Without major changes (unlikely), its likely the relationship ends - even having Wallis as a "secret" mistress is unlikely (the King is far to indiscreet because he either can't be or doesn't care to be) and its likely an ongoing source of embarrassment and speculation that will be unacceptable as part of any solution to the brewing crisis.Did anybody ever considered Edward to actually be talked into/ be convinced by Wallis Simpson to take her advice/offer of her to "give him up" (according to wiki third paragraph) and thereby avoiding his abdication at all ?
... maybe keeping her (at least for some time) as his - more or less - secret mistress ?
Wouldn't be the first time he had a 'lover'.
Aside from his 'traditional' unpolitical role, even if he m,ight have furthermore 'accepted' it he most likely would still produce the one or other ... 'verbal input' into public discussions on politics (similar to Kaiser Bill prior to WW 1).
And in being the King his words or opinions on politicians might have some effect on voters behavior and thereby in politicians behaviors.
What course might the british ship take with him present on the aft-deck ?