Do you really think that the French people would accept Elizabeth II as monarch? This would completely negate the idea of French republicanism, even if by this time Britain was long a constitutional monarchy. Yes, there are historical prescidents for royal unions between the two countries, but not anywhere near the modern period.
There is also the question of religion. The reigning monarch of Britain is the Defender of the Church of England. What would happen if she became the Queen of France? She would become the queen of a officially secular nation that is a predominately Catholic country. She would have a two options: drop her "defensor fidei" and disestablish the Church of England, or rule her French subjects as she rules her English Catholic subjects: as a minority group that enjoy religious liberty with the exclusion of certain rights, mainly the prohibition of royal succession. If she were to become the Queen of France, it might happen that one of her princes would like to take a French woman as queen. Besides running afoul of the Act of Succession, she would run afoul of French republicanism, Rome, as well as those in England who wouldn't stand for the disestablishment of the C of E.
I also doubt that the British and the French people would easily accede a non-democratic union of their two countries. If an Anglo-French union were put to a vote, I suggest that it would fail in both countries by huge margins.