Try taking a black cab from Heathrow airport to anywhere just a few miles away (basically anywhere which isn't a full fare to the centre of London). You will literally be screamed and sworn at.
I've used a black cab from Heathrow to central London a few times, but I still find using the Heathrow express more convenient and cheaper. Later getting a cab in Paddington to anywhere in central London is a breeze... compared to the M4/A4 constant traffic jams.
EDIT: I misread, this is my response when thinking they wanted to go TO Heathrow
Well then, that's a perfect example of a situation where Uber would be more appropriate. Black Cab drivers are meant for quick journeys through London. Why should they go well out of their way? It's their business.
(although, there's no need for people to scream and swear at you)