All BA passengers connnecting to or from BA long haul flights may travel on the Speedlink coach (LHR-LGW) free of charge! A free voucher is available at the BA flight connections desk/arrivals desk.

For more info on the Heathrow or Gatwick Express:
http://www.gatwickexpress.co.uk/
http://www.heathrowexpress.co.uk/

Bizjet: It looks like all options will work for you. Minimum connection time in Gatwick is 45 mins. so you should be ok. As Shareholder already stated your bags will be checked through so no need to pick them up at either LGW or LHR.

Good luck!