I know it's true for the Queen, but not sure about the Prince of Wales. His assets are tied up in the Duchy of Cornwall, aren't they?
If Britain becomes a republic, the Duchy of Cornwall reverts to Parliament, or no?
Charles is The Duke of Cornwall and has enjoyed a rather large income (in excess of 10 million pounds) from it each year which he pays taxes on. That income is his and would not revert to the government - not legally, anyway. His stewardship has lead the Duchy to become far more profitable that it ever has been. The Queen also recieves income from the Duchy of Lancaster as she is The Duke (not Duchess) of Lancaster.
The Duchy of Cornwall is by Charter of 1421 held by the Sovereign's eldest son and heir and is not inherited by his children if the holder dies before he inherits the throne. If the heir was a grand child of the monarch or a women they would not be hold the title.
The charter that governs the transmission of the Duchy has no provisions for inheritance by any other method than as the eldest son of the Sovereign. If the monarchy would cease during the Queen's life I doubt that Charles would cease to be the Duke of Cornwall. Without amendment to the Charter he could only cease to be the Duke of Cornwall by his own death or inheritance of the throne, by my reading of the charter.
Both the Duchy of Cornwall and the Duchy of Lancaster are governed by very different rules than any "mere" peerage. The Duchy of Lancaster is the personal property of the Queen and is inherited seperately from other Crown possessions. The Ducy of Lancaster, which is the Queen's private property, is valued in excess of 300 million pounds. I doubt any decendant of Queen Elizabeth II would need to rent a flat if the country were to become a republic. The Privy Purse funds are all derived from profits from The Duchy of Lancaster which are used to pay for the offical functions of numerous Royals (the Kents, et al).
The Duchy of Cornwall and The Duchy of Lancaster are not typical peerages nor are they Crown Property. They are actual landed estates the only true landed estates that still exist in England. Each is a "county palatine" and are uniquely held and maintained.
The personal wealth of each British monarch since 1399 has been largely the product of their status as The Duke of Lancaster (in addition to inherited wealth from those who were foreign royals). Many items purchased by monarchs using their personal wealth have been siezed and are now part of the Crown Properties (which includes artwork, stamps, jewels, land, ect...)
It's not like they can issue a law that strips them of all their money and property. We can't forget that the European Court recently ordered Greece to compensate King Konstantinos for property that they siezed following the fall of the monarchy there. The case that could arise in from the UK becoming a repuplic without reaching a settlement with the royal family could be massive. So, massive that it could bankrupt the new government. The Bavarian Royals still benefit from the settlement their family recieved when the Kingdom of Bavaria ceased to exist. A similar arragement would be in the best interest of any respectable nation that wouldn't take the route similar to that of Iran or China.
Additionally, I seriously doubt that they would be asked or forced to leave the country. I would imagine that it would be similar to how things worked in any of the 28 former monarchies that make up Germany. The families remain they are wealthy, respected, still hold many assests from their days as reigning families, work real jobs and still, generally, preform "royal" duties such as charity engagements.