It's a nonsense to suggest that BA "get priority" at EGLL. This is not some banana republic where such things happen with the major based operators, which you maybe have experience of elsewhere.
When there are periods of disruption BA are the FIRST to get cancellations, principally of their domestic services, even when others get no cancellations at all, and invariably end up with a greater percentage of cancellations at such times than anyone else.
From where I currently am there is quite a reasonable view from the ground of the traffic over NE London, and you do indeed see people getting a straight-in from time to time when quieter. I believe the runways at Heathrow from 0600 to 2200 are among the most saturated in the world. If you want something done about it maybe you can drop a note out of the DV window as you pass overhead Parliament when you finally get your approach. Alternatively, get your company to use Stansted - you can see it on the right hand side as you approach LAM. Nice and quiet.