I'm very certain that the British monarchy will survive because it's a matter of structure, not persons. We all focus too much on the persons, how they're dressed, what they do and say - but beneath that, there is a fundamental way of how a society organizes itself.
I think the more chaos elected politicians brew, the stronger the un-elected pillar of stability, monarchy as such (and not only the specific monarch), will be more appreciated.
After reading quite a number of books about the autocratic monarchies that fell in the aftermath of WWi, I have understood what seemed weird to me for years. Coming from a country were monarchy was abolished in 1918, it always seemed strange to me that Britons curtsy and identify so strongly with these rarefied, pampered creatures called royal family. I understand it better now.
The British monarchy, like the Scandinavian and Benelux monarchies, has survived because it remained flexible. I think the penny dropped when I read how horrified Czar Nicholas II was about the Norwegian crown, offered by the people to a Danish prince - and that prince accepted! Well, Nicholas II ended his life tragically and the Romanovs don't rule any more. But the descendants of the Danish prince still wear the crown offered to them by the people.
Evolution, not revolution, that's the secret of survival. I think nobody has a better sense of balance between stability and gentle adaptation than the royal families who are still on their thrones.
And the more unreliable and confused elected politicians look, the stronger and saner does the monarchy appear.
No no, Elizabeth II is intelligent, pragmatic and experienced enough to prepare the soil for the time without her. She knows her funeral is prepared in every detail, and Prince Charles seems to know very well that the role of Prince of Wales gives liberties a king doesn't have.
Can I judge it from far away? Maybe I'm wrong but I have the feeling the Britons won't give up their queen or king in order to get a president.
But but but. This whole contract between monarchy and people only works while the traditional motto of the Prince of Wales, Ich dien', I serve, is still the principle of the monarchy. It's not enough to be born royal - the royal has to earn the privilege of his/her birth to make the contract last. That may be the deeper reason, apart from simple snobbery, why for many generations royals married only royals. Not even high nobility was accepted. If you think of the way in which generations of younger princes accepted secondary roles, generations of princesses agreeed to marry unloved spouses for the good of the house... we can't and don't expect this kind of self-sacrifice any more, but all the luxury of royal life comes with a price tag.
As long as the BRF continues to pay that price, continues to show that it serves the people of Britain, I don't see a crisis on the horizon.
And as I said on the first page - Great Britain's monarchy-people-contract has survived the Hanoverians and Edward VIII... :-) it's certainly tough.