Which is exactly why they haven't meddled in it to this point. No British prime minister wants to be "the one" who started the ball rolling that ultimately ended the monarchy.
In Britain at least, the Queen's Consent (different from Royal Assent) is required for a bill to be debated in Parliament that affects (directly or by implication) the prerogative, hereditary revenues, or the personal property or interests of the Crown. Further, on rare occasions (such as for the House of Lords Act 1999), the consent of the Prince of Wales, as Prince and Great Steward of Scotland, or as Duke of Cornwall, must also be obtained where a Bill affects his interests. This is known as Prince's Consent.
I would imagine therefore that both the Queen's Consent and the Prince's Consent would be necessary before Britain could debate the succession. I wonder if both would give their consent, knowing the possible consequences?