I live very close to Heathrow as well, and I just don't take black cabs any more. They are there to get the "good fares" into the city and when they hear I just want to go about 3 miles, they have a snit-fit and the journey is just unpleasant.
I was told that it was because they have to wait in quite the queue to get an airport fare and then they can't come back into the queue again for like 3 hours. So if they only get a local short trip, they feel cheated.
So it is mini-cabs for us from now on. Fortunately, since we are local, we know a good company. We pre-book before we leave and then when we land, we call, and they are pretty good about being there within 5-10 minutes of when we come out of arrivals. Most importantly, they are happy for the business and it is a pleasant journey home.