There are a couple of ways to do this but they will be pricy......
1) HIre a driver to pick you up at Heathrow, drive you past the big sights in London and then on to Gatwick. Depending on traffic, this could be a problem.
2) Take the Heathrow Express to Paddington. Arrange to have a driver pick you up there, drive you around the main sites and drop you off at Victoria Station no later than about 2:00. Take the Gatwick Express to Gatwick Airport. (You might be able to do the London part in a taxi if the driver is willing to do this.)
3) Take a National Express coach from Heathrow to London Victoria Coach Station and then either a coach from there to Gatwick or walk a block to Victoria Rail Station for the Gatwick Express (You won't see a lot of London but this would be the cheapest option.)
There are no tours that pick up at Heathrow, visit London and then drive to Gatwick.
Personally, I think it's risky especially if your plane is late. I would just take the coach from LHR to LGW--which unfortunately doesn't go through London, and just chill at Gatwick.