Comment by ghaff
It's a pretty extensive system and the pretty new Elizabeth Line is great. But if you take something like the Piccadilly Line in from the airport, you probably shouldn't have a lot of luggage because a lot of stations just have stairs and platforms are often at a significant offset from the underground cars. (The double decker busses also work pretty well although they're not generally my default.)
The Piccadilly Line was opened in 1906 for gods sake, forgive them for not catering to people with 3 suitcases very well! That's part of the reason we built the Elizabeth Line, to enable a better transport option for people coming into heathrow.