I too would rather change airports the Uk side then the Turkish side, we have done both Antalya and Bodrum and whilst it was worth it to get to Turkey it is in my opinion far easier to land at Dalaman.
Easyjet's first flight out this year from Gatwick is either the 24th March or 27th, Thomas Cook's are not until beginning of April, (dont know about Stansted)I do know the Easyjet flight is pretty damn extorinate as I looked at changing mine but not prepared to pay that price..
Good luck in whatever you deceide to do to get there