Rig a referendum to remove it? (Queens has most of the support of the UK so it would need to be rigged for her to lose one).
There's no point replacing them with a modern, generic president because all they'll do is the same **** the Royals do, other than no-one would give a **** about any British president. No-one will give a **** about their family, whereas the Royal family as a wholes plays a big part in selling the UK as a business partner, charity events, official openings. The Queen makes no constitutional decisions in the running of the country, she prefers it like that, we prefer it like that. What will we gain by enforcing an elected guy to go around opening new shopping centres instead of a royal? The royals are icons. People like icons. You cannot manufacture an icon from some elected politician. Granted, Boris Johnson has some comedic character (to use him as an example of British political icons) but in no way could he and his family step into the shoes of the royals and make the same impact.
Getting rid of the monarchy is moronic. It would be the pass-time of a moronic republican who thinks it'll make all the difference, just to prove a point. This country would gain nothing from republicanism other than 'looking democratic'. Electing someone who shakes hands with our national allies and smiles is probably the biggest waste of voting.
We should instead be reforming our voting system including how we can pick and choose certain policies instead of parties. Parties get unpopular by doing stupid **** like the poll tax or the bedroom tax, regardless of whether they've encouraged business, aka jobs, to grow and be competitive. That's where our voting power really matters, in how this country is run, not because some poor ickle so and so is jealous they weren't born into royalty. If it was proven that cutting the royals would make a difference to the ordinary working person I can assure you it would have already been done.