Even though you have HEx tickets, I'd be tempted to tell a white lie. Tell whoever bought the tickets how grateful you are - then (if refundable) cash them in or (if not refundable) throw them away. You can get a train from nearby stations to either Blackfriars or Cannon Street. Change there (really easy at either of those stations) to the District and head to Barons Court and then the Piccadilly. Much simpler than the complicated faff of getting over to Paddington... If you're using electronic payment (oyster etc) for the journey to central London, the extra cost of using the tube to Heathrow will only be about a pound.