BA used to pay. However, there is still a trick that works in some areas, ask at IAD for the connecting bus ticket. Some offices still have them as part of the deal. The arrival at LGW is ground level, just to the right and in front of you is the bus ticket office. Just outside the main doors (glass area) is the set of bus stops. From memory, it is just to the right. The buses all start at south terminal and will then stop to pick you up at North Terminal. Remember to tell the driver which Heathrow terminal you need. From memory, it is T4 as I don't think that BOM was on the list of flights transfered from T4 to T1. The buses are every 10 minutes or so throughout the day. Beware, the road between the two airports can be an absolute nightmare at the moment in the mornings. Get out of LGW as soon as possible and get your wake up shower at LHR. The bus drivers know every back road between the two airports if necessary. Remember to through check yourself, if not your luggage these days. You are then in the system if the bus is late. First check-in is at the far right of T4 at LHR. If you are late, go to any customer service desk (like a check in desk but at the extreme left or right of the rows) and they will deal with you quickly.

A taxi from LGW takes about 5 minutes less and costs around $110. For that you do not own the car, merely borrow the driver and his vehicle for a hour. Welcome to the land of petrol at $6 a US gallon!

If you need any more, come back as I work at LGW!