I got a taxi to Bristol bus station, then a National Express direct to Gatwick. Wasn't cheap, but very convenient. Plus! They have TV and really comfy seats, so you get a few hours to relax before you get to your hotel and into the books. I think (?) was about £30 one way, plus about £20 for the taxi.
You can take a service bus from Cheddar to Bristol, but by the time you haul your cases around, you may as splash out on a taxi.
If you hit lucky you may be able to score a lift to BGS with someone. So hold off buying your tickets until you get to Bristol and see whats going on.