Why are they in London?
The BBC was made to move as it was too London centric.
Would we be better off with a building project somewhere broadly in the midlands, somewhere between Coventry and Nottingham and Brum where land is relatively cheap to build a world class accessible parliament in the centre of the country with 800 or 1,600 serviced flats available to MP's and PA's during their tenure.
Incidentally, I'm sure you all know I'm not based in the midlands so that's not just me having a Redditch bias or whatever. Just build something with good motorway, rail and air links. If it was close enough to the NEC it would have all that at a fraction of current costs and it would be a major building project and could even kick start a new town in the area.
Yes, it would probably then gravitationally pull lobbyists and dearer housing in to that area over time. Well, that's a boost to somewhere other than London then isn't it. Just make sure you understand the need for affordable housing and a living wage and that can be dealt with. It's too late for London, you can't wind back the price of houses in Peckham or Brixton now. But you could create a belt of not to be privatised social housing with green space - it might even be a template for other towns.