I don't know much about London transport as I am rarely there bar transiting at Heathrow so until more folks come on and post here is some info from the BA Dashboard which highlights some companies and might help. Addison Lee are there but I suspect Uber may be the cheapest option assuming there is not a price surge on.
LHR Transfers
More info here too
https://www.heathrow.com/transport-a...s-and-minicabs