I never thought about it but if you are staying at the LHR hotel and wish to lock into a fixed taxi price, then trying Addison Lee which is bookable via the internet would be an interesting option.

I find they are time and given they do a lot of pick-ups at LHRs, I am sure it will not be difficult for the firm to route a car to do a hotel pick-up prior to an arrival pick-up.

The website will give you a location to location quote on line and they stick to it. Pay by credit card to lock in the fare.