The protection and the shooting of any wild bird is covered by the Wildlife and Countryside Act 1981. A general licence is required to shoot wood pigeons which may be allowed to stop damage or disease. See click here see schedule 2. There are wildlife inspectors. Suggest you contact DEFRA click here
I believe that it is illegal to discharge any firearm within 25m of another propery or public area so if he was within that distance of the property on which he was shooting then it would be illegal to shoot anything. As Woolwell has stated, there is also a safety issue - the pellets which miss do not just stop where the target is but continue for some distance and are still moving rapidly as they fall.