Uh, Chamberlain was prime minister at the time. And, at the time, Britain was hoping to avoid war and Chamberlain seemed to have pulled it off. It would have been political if George had invited a politician running for office onto the balcony.
Actually there was criticism at the time - Chamberlain's peace deal was highly political and there were those who were utterly opposed to appeasement - the King and Queen were both ardent appeasers (like many Briton's at the time who dreaded another European war) - it was a show of political support for a Prime Minister both the King and Queen liked personally and as such was touching on being unconstitutional as the Monarch is supposed to be above politics even if many of their subjects were equally overjoyed with the Munich agreement (Chamberlain once described Queen Elizabeth as the "only royalty" he felt comfortable with and liked and he was also fiercely anti the Duke of Windsor and was appalled when he discovered how the Duke had hoodwinked his brother over the financial settlement)
Edward VIII was by all accounts less bothered about the work that went with being sovereign - he talked the talk and was aided in this by the press (especially Lord Beaverbrook) - the something must be done after visiting miners in Wales for example (though they didn't report his dinner that evening where Wallis was wearing yet another stunning piece of jewellery which when admired by a guest the King was heard to say I approve of splendour) - to be fair he found the whole business of being King boring.
His only option from Baldwin and by enlarge most of the commonwealth countries (with some exceptions) was to remain on the throne as a single man or abdicate - even those who supported him urged him to stay and get the coronation over with before looking for a more long-term solution to his love for Mrs Simpson.
Edward simply was not rational about the issue - even proposing at one point to simply leave the country to give everyone "time".
If he insists on remaining and marrying then Baldwin would resign and he had already guaranteed that most of Parliament would not form a new Government - Edward's parliamentary support was minimal even if someone like Churchill might have been willing to serve it is likely that an election would have been called and a King's party formed from all parties. In other words a divisive election campaign that was effectively about who the King could marry. There was support among ordinary people for the marriage and it is very debatable whether a King's party say lead by Churchill with the support of some elements of the press might have won a majority.
That still leaves the commonwealth realms with a problem but most might have come round (the Irish didn't care and in OTL used the abdication as an excuse to remove the royal connection almost completely from the Free State's affairs - eventually moving to a republic in the late 40s)
In short it would have been a mess - Edward VIII was not especially pro-Nazi but he was pro-German and anti-war - his fondness for Germany dated to a visit shortly before the first world war and he got on well with some of his German cousins, his anti-war views were largely formed by visits to France during the First World War and the huge loss of life ( he was not alone in those views). Wallis and he both had so much resentment about their later treatment that is is easy to read into their actions during the War and afterwards that they were virulently pro-Nazi but I don't really believe either were quite that intelligent and Edward's vanity allowed the couple to get involved with a range of flatterers from very dodgy backgrounds.
Interestingly the one politician who was fond of Edward (at this period by the Second World War he was done with him) Churchill was absolutely opposed to Hitler which would have been interesting.