You are best to speak to someone direct on the phone to make your reservation. I doubt you can make this booking on the internet.
Cyprus turkish airlines (
www.kthy.net) fly from stansted and Heathrow, not sure about Gatwick. Some of their flights let animals on board. They used to just charge as excess baggage weight. Not sure if that still applies though.
Friends of our brought their dog out with thomascook 4 years ago from gatwick, but it did work out considerably more expensive.
www.flythomascook.co.ukYou must speak to defra who will give you a list of all the necessary documentation amd inoculation certificates from your vet that you will need. The will also give advise on the size of travelling box you will need to buy for your dog.