They might both be deposed shortly, either by Britons themselves, Muslims or the E.U... by their own inactions...
The Monarchy remains extremely popular in Britain, so the British people are unlikely to do it. The EU doesn't have the power (although at some stage, if Britain keeps giving it more and more, it might), and while there are certainly problems with militant Islam in Britain, there's a world of difference between being able to set off nailbombs on the Tube, and overthrowing a government.
Britain endured terrorist attack after terrorist attack after terrorist attack from Irish Republican terrorists. It just deals with them when it has to. The Islamists are still rank amateurs in comparison.