Apologies if this has already been considered, but would the Kingsway tram subway in central London work? (more about it
here) It's not an operational railway so may be less hoops to jump through than using somewhere that's on the 'real' railway.
It has been used to portray an underground station in a film - I've not seen the film so don't know how realistic it was, but might work for you (I'm not at all expert in these things, but presume you can play tricks with camera angles that might make it more convincing.) I know the London Transport Museum arrange guided tours down there, but don't know if the bit of TFL you have contacted would have thought of that.
I presume you're probably after the bit with tracks in, rather than the station building, but just another possible tangent - there's pedestrian tunnels under the thames at Woolwich and Greenwich which have entrance buildings with a lift (which sometimes works) and a spiral staircase, so again with the right camera angles and a bit of fudging, what's inside these might serve the purpose as part of an Underground station if you end up filming the railway bit outside London. There's some pictures of the Woolwich one
here ('murky depths' local news blog / site) - the ownership of them's a bit complicated, as the south end of each is in Greenwich borough, the north ends of the two are in Newham and Tower Hamlets respectively, but Greenwich may be the best first contact.