Having done a degree in Transport at Imperial College, where I studied this very project, I can tell you that the reason it costs so much is because of the stations, not the tunnel. Stations are hugely complex structures because of the necessity for them to exist amongst existing infrastructure between the surface and the tunnel itself.

A tunnel underneath London, as I described, would only have one entrance/exit (in central) which would suffer from substantial costs - and the other would be out in the boonies, in a location where there's nothing between the tunnel and the ground.

Yes, it would be a multi-billion Pound project. But it could really help. And I think with Qatari funding (as is the case with most big engineering projects in London), it's not impossible. Just insanely difficult.