Given that the H'Ex tickets and taxi fare will set you back GBP 32 if not more, I'd be more inclined to arrange for a car service from LHR. It'll work out to be about the same (or even less).
Is there still a problem with getting taxis at Paddington as Jim90068 alluded to? I heard some time ago that the traffic flow had been changed much to the taxi drivers' annoyance and they decided to avoid looking for fares at the station resulting in long taxi queues.
I'm sure someone in the Airports > London Travel > London/U.K. or failing those Miles > BA forum could recommend a good car service.