Outside domestics, the major markets all have competition. It's a very British thing to have this "second force" mantra, passed from BUA, to BCAL, Laker and now Virgin. The real competitors are much larger carriers like Lufthansa, Air France and KLM in Europe STAR and Skyteam respectively, and up against Emirates, Etihad et al to the Middle East with the traditional long haul carriers still on the same routes. Cathay, Singapore etc. As for competition to the US, LHR has gone from American and United vs BA and Virgin under Bermuda 2, to Continental moving in and merging to make a stronger United, Delta moving in and adding capacity and quality onto LHR-JFK and maintaining what Northwest had started to US Airways moving in on LHR-PHL. In short, since 2008 LHR has seen the addition of :
CO LHR-EWR, IAH, CLE (since suspended)
NW LHR-MSP, DTW, SEA (since suspended) with DL maintaining MSP and DTW
US LHR-PHL
DL LHR-ATL, JFK, huge ramp up on LHR-JFK on the B764.
In BA's favour, they mow have ATI with American but you can't say LHR-US is a BA monopoly, far from it. It is the US competition added to BA raising it's game added to Virgin not being in an alliance that is making Virgin underperform in comparison. The market is incredibly competitive, it is VS who are not....