I believe the BA promotion (back when they had many LHR-LGW connections) may have disappeared now. The schedules and prices are at http://www.nationalexpress.co.uk/ and you can buy e-tickets, tickets at the counter, or (cash) tickets from the driver.

If you want to look into an MPV taxi, to see how much it costs, ask either Airport Direct or Addison Lee for a quote. I tend to use the former for price, but the latter have a better service.

Don't get a taxi off the rank. It'll cost you a bloody fortune.