Another option is 'The Airline' coach running from Oxford to Gatwick via Heathrow. If you download the Oxford Bus app, you pay £20 for a flexible ticket between LHR and LGW. Activate each segment in your phone as you are about to get on the coach. (Both segments being valid for 12 months from purchase). Plenty of space for cases under the coach if you have them. You can use in either direction after purchase (as in: you don't need to specify the date, time or initial direction when you cough up your 20 notes, as long as it is a return ticket).