The Queen does have personal wealth and property, and she would be able to keep them. Some other members of the family, such as Prince William and Prince Harry, also have their own wealth and own property, while many members of the extended family already actually hold down -- surprise! -- real jobs.
Such properties as Buckingham Palace, Windsor Castle, and St James Palace aren't owned by the monarch anyway, so they would probably be run as museums by the state.
Those members of the family that are currently dependent on Parliamentary Annuities in return for carrying out royal engagements (those annuities are reimbursed to the state by the Queen) might be given settlements of some sort, since some of them would be too old to establish careers. And there might be other settlements made to compensate the family for the loss of revenues from certain duchies, those of Lancaster and Cornwall specifically.
The negotiations would be complicated, but ultimately, the former royals would probably simply settle down as wealthy Brits, with some former glory still clinging to them. It's highly unlikely that they would be forced into exile as members of defunct monarchies in other countries have been.
The abolishing of the monarchy would raise serious questions about British titles in generals. Would it also mean the disappearance of the titled aristocracy, people who would suddenly become "Mr," "Mrs," and "Ms"? Or would the titled Brits, royal and aristocratic, continue to use their titles as many Europeans do, even though these would no longer have any real meaning or substance?
It's an interesting question.
Posted 863 days ago