All the UK airports require pets to come in as cargo, but not all airports allow pets and have checkpoints for them to be screened.
Personally, Based on my experience and airlines statistics for transporting animals, I would only fly a pet cargo with British Airways, and only at moderate temperature times of year. When I wanted to fly into the UK from America in winter, i would fly to France and then take the ferry across to England.
You might want to look into taking the ferry across to the continent and then flying your dog in cabin with you to Cyprus. It may be cheaper too.
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