Its surprising Edward VIII was allowed to die of old age in France.
Not surprising at all.
Leaders who are overthrown have a good chance of being killed. Ones that abdicated, not so much.
Plenty of examples of both overthrown and abdicated living.
Charles V of the Habsburg empire
Phillip V of Spain
Charles IV of Spain
Napoleon III, France
Manuel II, Portugal
Pedro I and II, Brazil
These are just the ones that immediately come to mind. There's a few residing in my foggy memory bank that I can't pull out right now, but I'm sure if I put some effort in, I could find them, and other examples. Ditto for dictators.
Violent ends are not uncommon, but neither are exiles.
The British Monarchy of 1930s was not as ceremonial as today, but it also wasn't truly running the empire. There was little need of bumping off Edward III, and most likely would have been massively counter productive.
As to the OP, this was the era of execution, when it was acceptable to settle a threat with an axe. A republic (especially if it had any struggles) would be at extreme risk from a monarch in chains. It might be easier to re-ignite monarchial sentiment if a deposed King were still alive. Whether OTL republic would collapse quicker, or hold on a little longer, probably doesn't make much difference overall. The end result is likely the same.