Best option depends on who you ask... There's some space for bags by certain doors on Piccadilly Line trains, see
https://en.wikipedia.org/wiki/File:P...eLIneTrain.JPG for what that looks like. Personally I've no problems with 2 suitcases on the tube outside of rush hour, but it depends what you fancy! Likely one of you will have to stand by them to stop them falling over though, unless you're able to do something cunning with lying them down with them not moving
Tubes start at Heathrow, so getting space for you and bags isn't as bad as when you start in London going to Heathrow
Otherwise, quite a bit of space on the Heathrow Connect (slower cheaper stopping train service to Paddington), then the sub-surface tube trains from Paddington to St Pancras are quite a bit bigger