I've done it twice. I flew United Airlines to Heathrow then, because I'm a total cheapskate, I caught the bus to Gatwick to fly Air Baltic.
Anyway I wouldn't recommend it to a novice traveller unless you had a good reason. The National Express bus terminal was difficult to find, signage was poor and the airport staff were unfriendly and flippant with help. Lots of different buses for all over London leave from there. Which bus goes to Gatwick? Even when you get to Gatwick, which terminal do you get off at? The driver yells out "North Terminal!".. uh.. do I get off here? Which terminal does my plane leave from? If you pick the wrong terminal, what then? I constantly checked my phone to see if I was going to miss my connecting flight. It takes at least 90 minutes for the drive. What if there was a traffic jam? I'd make sure I had at least 5-6 hours before attempting it. You need to go through immigration and customs twice then re-check your bags. That's going to take at least an hour and more of your time.
If you're going to do it - you won't have internet so print everything. Maps of Heathrow terminal, bus numbers, bus terminal location, terminal of your connecting flight, terminal of your arriving flight.You'll need British pounds cash for the visa and the bus. They put your luggage under the bus, so I sat on the left then I'd see if anyone tried to steal my bag.