Just playing devils advocate with the above statement, the BE EDI-LHR was feeding the same demographic as BA were. They had numerous codeshares to feed onwards to long haul operators, including Virgin Atlantic, Cathay, Air India, Etihad, Emirates, Delta, Singapore etc etc. The problem I suspect with BE flying domestically into LHR, and every other operator who has tried to go up against BA, is that BA can afford to take a loss on the regional market as it supports the long haul one.
This is the same problem worldwide. Most regional airlines that fly into major hubs are supported by larger flag carriers who can afford to take the hit as it keeps business flowing to the more profitable side of the company. Think American Eagle, United Express, Delta Connection, KLM Cityhopper, Lufthansa Cityline, Alitalia Cityliner, TAP Express, etc