I checked with BA. They confirmed that my luggage will be checked to the USA and I will not need to collect it for the overnight connection. That's good. Without bags, all options are on the table still.

I don't care about attitude from taxis. I understand them, but it comes with the job. I have parts of my work that I hate too. We all have to deal with it. That said, I think ordering a car in advance or Uber would be cheaper anyway. Perhaps we will try the bus or tube to the hotel in the evening and then decide if we order a car for the return in the morning depending on how it is?