You actually have quite a few choices, but I think 3 hrs is tight for any of these methods. LGW is 30 minutes south of London's central district even with an express train service. Let me get this map out....
# 1- Express Bus "Heathrow Express" from LGW to LHR nonstop. This runs I believe every 15 or 30 minutes. If traffic is light, its probably the fastest method. But the M25 roadway is probably always busy so you could end up stuck in traffic. I think it was around 30 US dollars.
# 2- Express Train- "Gatwick Express". This is heavily advertised as you are exiting LGW, and is the fastest method by far for train service. It runs nonstop from LGW to Victoria Station in central London. The fare is 12 euros one-way and takes 30 minutes, with trains leaving every 15 minutes. At Victoria, purchase an Underground (tube) ticket and take the Yellow line (Circle Line) to the South Kensington station. Transfer at South Kensington from Yellow to Blue line (Picadilly Line). Take Picadilly westbound all the way to LHR, which has 2 stops, T1 thru 3 and T4.
# 3- Slow multistop commuter train. "ThamesLink" This runs from LGW to central London and the fare is 10 euros. You would have to connect from Thameslink to the Tube at Blackfriars station.
I have used all 3 methods and would use the Express Bus or Train again when I go back.
FYI, The Tube is the Underground (subway) system, ThamesLink/NationalRail is a slower multistop train system, and Gatwick Express is just a high-speed nonstop train from Victoria to LGW. They may share common stations, but you'll have to purchase separate tickets to transfer between the operators.