I would suggest that the current King could be impeached by Parliment (OK, so it has never happened but it is possible under the agreement that Willaim and Mary signed with parliment after the Glorious Revolution) and replaced with the heir apparent.
This might be a problem if the heir apparent is very young. Possibly you could get another Glorious Revolution with another foreign King.
I still feel that a popular uprising NOT supported by the majority of Parliment would fail if for other reason that weapons were not readily available (except farm impliments and whilst these are dangerous they are not designed for street fighting or pitch battles).
It's possible you could get a situation where the Kings and the Lords are on one side, the Commons is split, and the People are angry.
How does the availability of weapons contrast with the French Revolution?