Check out the trip reports (just google it) on this particular flight -- the aircraft is supposed to be equipped with GA's newest hard product. I want to fly in F on this route sooooo badly. Let us know how it goes!
I believe there's a direct bus you can take from LHR to LGW. The express trains are unreasonably expensive.
Depending on the time of day and the traffic, another way to get to Gatwick is taking a local bus from either the bus station that serves T1-2-3 or one of the stops in front of T4 to Feltham Station (only about a 10 minute ride from T4), then taking a train in the direction of London Waterloo for about 25-30 minutes, getting off at Clapham Junction, and transferring to the Southern Railway's services on the London-Brighton line that stop at Gatwick's railway station. Usually, their semi-express services stop only at East Croydon before reaching LGW. The ride is less than 30 minutes from CJ to LGW.
The problem with the Heathrow Express is that it terminates at Paddington, and the Gatwick Express leaves from Victoria, meaning that you have to navigate a 20-30 minute trip on the Underground.
Another alternative, although it would take you around 2 hours to do, is to take the RailAir bus to Reading Station, and then take a direct train (I think the carrier is First Great Western) from Reading Station to LGW Station. I've taken this train and it takes around 70 minutes via Guildford.
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There used to be direct bus service between the airports. In fact, the bus from Cambridge was frequent and went to not only LHR and LGW, but also one other airport closer to Cambridge, but I forget whether it was Luton or Stanstead. I would expect the same setup from other places so that there would be lots of buses between LHR and LGW.
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If you are going between Heathrow and Gatwick the quickest way is just to get one of the National Express busses between the 2 airports... cost is 25 pounds whcih will be cheaper than going into central london and getting the train out to LGW. The ride takes about 50 minutes and runs every 10 minutes or so.