I've never had a problem with London Bridge cabs, but I have to say I'm getting more in to using Addison Lee, they're not too expensive, their drivers know their way around London, the cars are really nice and they text you all the info once the car is on its way to you.
Great service if you ask me.
Express (403 3333) are good, but you need to open an account (not problematic - takes about 5 mins on the phone).
They've taken my wife to and from places early and late and she's always had very good reports for the drivers. They've also done a couple of airport runs for both of us, and have been good (and not charged waiting time the once we were delayed)
They're probably not the cheapest, but the cars are clean, new and comfy (which is worth a bit extra for me, as I so seldom go in a car and when I do it's our old banger, so it's a bit of a treat when I get a cab - which is probably a bit sad to admit)
I use cabs fairly frequently for work. (i have to carry lots of equipment but do not drive).
I generally use waterloo cars 7401 3333, they are pretty reliable and fairly cheap too. If for any reason they can't take my booking I use express cars (number above) which are more expensive but do have smarter cars ( not that waterloo cars are old bangers). You do not have to have an account with them but at busy times eg around Christmas time account customers take priority (naturally) so you may have to wait a litle longer for a car if you are not booking in advance.
Between the two firms I never seem to have a problem getting a car when I need one and their numbers are so similar its easy to remember them.
I have had an account with Express for quite a long time.
Good news about being an account holder is that you get priority on tube strike days and at busy periods (good).
Just check out with them the difference between paying for the journey on your credit card account vs. paying cash for the journey...
I fell out with their booking operator over this subject (cash is much cheaper than account credit card) - and they refuse to let you change method of payment mid-journey on the way to destination (inflexible) - so they lost the return journey job as a result (round trip journey had been booked at outset) - so from an admin point of view cancelling a pre-booked return journey is easy for them (?)
Beware all, and remember to ask them to quote for credit card vs. cash price when you next order a car from them using your account!