We used the bus to get into London some years ago. It was easy and pleasant, and nice to up above ground. Back then (1998) it was a double-decker and we rode in front on the upper level for a great view. I don't remember what we did with luggage but it was not a problem.
These days, we use the 'Duo Saver" ticket for two traveling together to get a good price on the Heathrow Express. It only works if you are also returning to Heathrow to fly out, as it is a return ticket. £50 for two for a round-trip ticket ( return must be within a month), making it £12.50 pp each way. More than the Tube, but much less than full,price, and very comfortable and easy. At Paddington we take a cab to our hotel unless we have booked one right there (Hotel Indigo).