T5 only has 4 lifts to/from the Heathrow Express/Underground platforms, so I am assuming they also serve the car parks. They run on a pattern meaning they go straight from the bottom to the top and then stop at intermediate levels on the way down, or something like that. They have no call buttons anyway. There are escalators from the trains, but it takes a while due to the sheer height of the building.
I am not sure if T2 will eventually have it's own baggage system (and a train to connect the satellite piers, at the moment it's a long walk) but there is space to put one in under the building itself. I know they will expand T2 sideways over what is now T1, so perhaps when that happens, a bigger baggage hall will be built and the current T1 system will be part of that?