Recently there has been some confusion about this in the media.
The grey squirrel is regarded as an invasive non-native species following its inclusion under Schedule 9 of the Wildlife and Country Act 1981. Under Schedule 9 of the WCA, it is illegal to release a grey squirrel into the wild, or allow one to escape, without a license granted by Natural England. This means if you trap one, you are obliged to humanely dispatch it.
Confusion has arisen because all wild mammals are listed on Wild Mammals (Protection) Act 1996, which some people take as meaning they cannot be killed. In fact, all this Act does is legislate the manner in which they can be killed – i.e. it is an offense to cause them “unnecessary suffering”. In short, caught Greys can be killed, but they must be killed humanly.