I can't, for the life of me, think of a single reason for the Royal Family to have any presence in the British government.
Te Queen might have influential power, but the real power lay with the PM. At any rate in theory it is her government.
I think they could continue to exist as a sort of curiosity, an attraction for the tourists, the same as any other pop idols, (Lord knows, there's plenty of Americans who can't get enough of gazing at that bunch of stiffs on a balcony.)...
I will not deny there are a large number of Americans who're fascinated by the British Royals, but that's because the Royals belong to the British and are a sort of wondrous curiosity. But I think you miss the most noticeable or important aspect of the British Royals....it's the British people's heritage. True, with more and more Britons getting higher education, they might look at the Monarchy with renewed eyes, and say it's not as important as it once was; and they would be right, a lot of the brilliancy of this institution has faded, especially for the younger generation, but still psychologically, it does play a meaningful role for the older generation.
I was especially captivated by the ups and downs of Prince Charles, Diana and Camilla; they were so entertaining. And I was shocked by the untimely death of Diana. When Prince Philip, and Queen Elizabeth go to the hospital it's all over the media and we do keep up with what's happening.
I bought a dvd of the Royals entitled "The Queen." (2006) starring
Helen Mirren as Queen Elizabeth II, and Michael Sheen as PM Tony Blair.