The above info about lounges is all correct.

However I might suggest this as an itinerary, depending on how much luggage you have (and how much desire to carry it around):

High-schlepping-tolerance: Arrive LHR, go to the arrivals lounge for shower, breakfast, etc. Then proceed into central London by Tube, stopping in the centre for something you want to do - walk around Covent Garden, go to a museum, shop on Oxford Street, whatever takes your fancy. Proceed to Victoria station by Tube and take Gatwick Express to Gatwick. This should get you 3-4 hours in central London.

Possibly other option if more luggage than carrying desire: Arrive LHR, go to lounge as above. Transfer via easiest route (probably National Express) to Gatwick and check in your bags.Then take Gatwick Express back to London to spend some time in central London, then go back to Gatwick and depart on your flight.

Hopefully someone else will comment if the Gatwick early bag re-check scenario is plausible.