I'm afraid I can't see the monarchy being done away with for some time, whatever some here would like.
This programme was an attempt to look at what a monarch might choose to do under their constitutional rights as monarch. Whether or not Charlie decides to be called Charles III or Alfred II or whatever is irrelevant to the story. After all, it is 'before the event' fiction.
By the way, I do find some people's antagonism to the monarchy on the grounds that it doesn't represent them quite amusing. The only part of the UK that has never had a national representative on the throne is N.Ireland, as far as I am aware. In fact, it could be argued that both Scotland and Wales have have representative monarchs since the last English representative.