How could the British monarchy be dismantled?

I'm not convinved that it is an anachronism. Whilst I am fully behind democracy, the concept of an embodiment of the state is quite a useful method of getting stuff done. An elected position is another way of doing it but the problem with that is that they would have more power since they have a mandate so then would you have a Prime Minister? If you have both they would interfere with each other so I don't think that works. I suppose you could have a representative from Parliament have that role like the Speaker of the Commons. The monarch has benefit of bringing continuity to things so the PM of the day does talk to the monarch on regular basis and in a way has to explain themselves. The monarch can offer advice having been around longer and familiar with the issues. You wouldn't see that with an elected official.