A while back I too was wondering what happened to the check-in area. At the end of it's life only *A airlines were using it. But searching through threads it looks like it was BAA and not any particular *A airline that made the final decision to scrap this.
Apparently BAA owned the area where the check-in desks were located and decided it'd be more profitable to turn it into shops

Anyone know if these shops are part of worldpoints?
I think self-service check in desks at Paddington are mostly useless, does it really save much time using a kiosk at the station, it's bulky luggage that you want to dispose of at the earliest possible opportunity.
When the check-in system was in operation, what happened to the luggage at the Heathrow end, was it all unloaded at Heathrow Central or was baggage for T4 stored separately, alternatively was there a luggage drop off point the train went to after dropping off pax? Did any SkyTeam airlines ever participate in this scheme?