There is a Concorde parked permanently at LHR, on a remote stand, near the perimeter fence, pretty much opposite Hatton Cross tube station.
It was moved there not long ago, as it used to sit somewhere else, but it's not moved under it's own power for a very very long time. They did, apparently, repaint this aircraft though, which is a nice touch as she was getting a bit rusty and dirty just sitting there. Quite what they will do with her is not yet known, even though she's not flown now for nearly 10 years!
I think at one point they wanted to put one of the aircraft inside T5 as a display...or at least take the Concorde model which sat on the roundabout at the entrance to T1,2,3 and use that in T5....However neither plan came to fruition and now a giant EK A380 sits on the roundabout and I have no idea where the Concorde model went!