I wouldn't be blindly trusting in asking my hotel or B&B to arrange a cab to Heathrow for me. I'd ask some questions. I've been burned; fortunately not on as big a trip as from Bloomsbury to Heathrow, but enough to remember for a long time.
A few years ago I had a very good week-long stay in a small family-run hotel in Bloomsbury. On my last day I had to get to Paddington for a train and they kindly offered to call a taxi for me.
It turned out to be not a black cab, but a guy in a regular car. He was rude and unhelpful with my bags, even grouchy about opening the trunk of his sedan for me to put my bag into it. (London's regular black cabs are vehicles with space in the passenger compartment for bags; that sedan didn't have that kind of space, just a regular back seat in a small sedan.)
At the end of the ride I paid him a lot more than I'd paid for my taxi ride to the hotel from Paddington. My ride to the hotel had been in a black cab that I'd boarded in the taxi rank at the station. I don't know if the guy my hotel called for me was a licensed minicab operator, or just some random friend or relative of the hotel folks. I am certain that they got a commission on that "taxi" ride.
Here's some info from Transport for London about taxi and car services: https://tfl.gov.uk/modes/taxis-and-minicabs/book-a-taxi