Heathrow Harry:
What surprises me is that ATC at one of the worlds best and major airports have no way of calling up an overlay with major obstructions on it on their screens
I'm sure we have that at LHR ... don't we???
That is a question for the staff at Heathrow.
In the U.S. almost all of our TRACONS have an Emergency Obstruction Video Map (EOVM), which they can call up very quickly if need be. I have linked a PDF file for KLAX. The first page is a portion of the MVA video map, the second page is the same area, but with the EOVM video map.
The EOVM provides only 300 feet of vertical clearance.
Index of /LAX