JB gives excellent and comprehensive advice as always, but I have a little experience with travelling with a wheelchair to add.
My wife uses an electrically powered chair, but can walk short distances. Yes, you will be able to get through the formalities at Heathrow faster, but you will be the last to leave the plane and if, as happened to us earlier this year, you are not parked on an air bridge you may have to wait some time to be lifted off.
Whether to stay at Heathrow or go straight to Southampton is a case of deciding which is the more daunting: Tackling a 1½ hour drive in a taxi or 2 hours on a coach to get to a Southampton hotel, or having to stay the night and up sticks and travel to Southampton in the morning.
I know which my wife would choose, although it might depend on the arrival time of your flight. Being met by a helpful pre-booked driver and driven to the door of a Southampton hotel would, in our opinion be well worth the cost. You would then, as JB says, have time for a leisurely breakfast and look around Southampton before boarding your ship. Southampton, in common with most of the UK has come a long way towards being wheelchair-friendly in recent years. Dropped kerbs at most crossings and level access to most buildings are the norm.