But there are fast and frequent train services that cost less than that, so it doesn't make a huge difference whether or not Uber is an option for that route.

All of which are completely unsuited to use with luggage, and/or require you to first get to Paddington station.

So get the AL to Paddington then get on the Heathrow Express. The Heathrow Express has never been very crowded in my experience, and has at least as much room for luggage as a cab does.