Addison Lee seems to be involved in anything to do with wheeled vehicles, from couriers and movers to cabs and even motorcycle cabs. I've used them before for early morning rides to the airport and they are quite professional, arriving a few minutes early and calling your mobile to let you know that they've arrived.
They seem to be pretty dominant in London, so you can usually count on being able to book a car, but at the same time because the operation is apparently so massive you need to get used to the rules and not expect them to cut you any breaks - there are fees for being late to the car for instance. I've also found that the drivers tend not to be as personable as many taxi drivers - although this may well be because at 6am nobody is terribly chatty...
Still, it's nice to be able to go door-to-door to the airport and having the payment billed to you helps for when you are arriving from international flights and are short on pounds.