Airport change Heathrow to Gatwick

You could but it would be quick hit. The two airport are like 5 and 9 oclock roughly. You could take the train into Victoria station from Gatwick and then the train our to Heathrow. From Victoriait is about an hour ride and one change via the tube. Or you could go to Paddington and the Paddington Express from there. You could have about two hours or a bit more around Victoria which is near the Palace.

Does the six hours include time to clear immigration and customs on the way in (say an hour) and the time to check-in and clear security on the way out (say two hours)? Because if so the three hours you have left will be largely taken up by travel. It's an hour on the tube from Heathrow, so you'd need to spring for the expensive Heathrow Express (15 minutes, I think?), and then you have to take a train to Gatwick (30 minutes?). Add in lunch and transit time within London and I don't think it's worth the hassle. Have a leisurely English breakfast at Heathrow and take the National Express coach over to Gatwick. And don't forget your inbound flight may be late...

I wouldn't chance it either, with such a short time. There's no telling how many other planes will be arriving at the same time as yours, and passport control be very slow if there are several. And you need to allow adequate time to get checked in, get through security, and get to your gate for your outgoing flight. And what do you intend to do with your bags while you are in transit? If you stop off in London, you'll have to drag them around with you.

I'd agree that six hours is not enough time to do anything interesting without being rushed, and that for those two airports the coach transfer is more convenient than going via central London.