The King marries Wallis without the support of Parliament, his Prime Minister, the Liberal and Labour Party, Trade Unions, the Dominions, the Upper and Middle Class, and a large portion of the Church of England, all of whom drew the line in the sand.
Yes, the Government would resign, both that of Baldwin and the Dominions. Attlee and Sinclair would refuse to form a Government, and a snap election will have to be called. The Public will be divided along class lines, the Working Class in support, whilst those in the Upper and Middle would oppose the King's choice. The General Election is unlikely to go well for the Conservatives, and will likely swing to Labour, although it's possible they would simply refuse to form a Government, and the King asks a minor pro-Marriage Party to form the next Government. In the Dominions, you see a similar effect, with the result being constitutional and social chaos across the board. Whilst certainly not catastrophic, you will still see a level of chaos.