I guess it depends on the context in Britain proper. If it happens around the near term, then I would think that the idea earlier about the President inheriting the status of monarch, like the Andorran example, would be the right fudge.
However in a different Britain, say with the rise of a popular movement, maybe not. Say that Britain has a wave of civil unrest, which gives rise to a popular movement, not necessarily revolutionary, this could easily spread to the dependencies and then end up with some sort of wider settlement, where the local people vote for a change in status. Annexation and federalisation or the like.