Two very important points, 1) Commonwealth has always been the tradition British Republican name (well not that important but it just has) and 2) Super-Patriotic Ulster would go bat-shit crazy on hearing the Motherland had dumped the Monarchy. Also remember, Northern Ireland has devolved rule from 1922 into the late 60s or early 70s so that pose even more problems.
Edward VIII continuing to rule, regadless of the whole Wallis Simpson thing (say he settles with some other girl) is bound to be tense at times. For one his supposed Fascist leanings will be much more in the spot light, perhaps he fails in his duty as leader of the nation during WWII, leaving politicans like Chruchill to take all the glory (this would no doubt deflate royalist pride). Also he lived into the 1970's, meaning no great pomp and ceremony Elizabeth II coronation in 1952, which did a lot to boost British pride and hopes for a "Second Elizabethan Age".
He was also known for his interfering nature as Prince of Wales, and briefly as King, if any modern British monarch is likely to bar legislation (which most agree would be the death of the monarchy) it would probably be Eddie here.
I could see him becoming something of an embarrassment during the Swinging Sixties for Britain's image. No young Queen with hip sisters, just an ancient King, totally out of touch with the 'New Britain', perhaps the odd gawfaw regarding the colonies and racism. Imagine if Queen Elizabeth, soveriegn of a good chunk of the globe, instead of harmless, moronic Prince Philip, was the one calling the Chinese slanty-eyes etc. Add to all that his less than perfect image with the older generations over his connections (however tedious) to Fascism and virtual none-existance during WWII (perhaps he moved himself to Balmoral 'for safety', meaning no "were all Londoners" schtick).
In such an enviroment, an given the right moment, say he vetos legislation regarding a Labour government's taxation on higher earners in the late 1960s (the maximum wage stuff they tried to introduce was absolutely potty with hindsight, but such move at the time Edward would soon see himself painted as one of wealthy elite VS the democratically elected government).
The act outrages Parliament, quickly moving the issue beyond the legislation and leading to demonstrations outside Buckingham Palace. After a few days of tensions Eddie might apologise and offer the removal of all Monarchal Powers, handing them over to the House of Lords. However such a move would quickly be seized on by Republicans, calling for a Head of State with power the people can trust, or at least control, ie a President via elections.
And so under massive pressure, King Edward VIII gives Royal Ascent to the 1969 Commonwealth of Great Britain and Northern Ireland Act, removing the House of Windsor from power.
Using Edward VIII in a semi-normal TL, thats about as early I see a British Republic coming to power.