All wild animals are protected to some degree by the Wildlife and Countryside act.

All wild birds are protected and can only be shot where General licences give that authority / exemption.
Bird species that can be shot under a General Licence, and the reasons, will differ between England, Wales, Scotland and Northern Ireland. As such, a shooter needs to make sure he is working to the GL in the area they are shooting in.
As above, birds can only be killed, where all other non lethal methods have either been exhausted, or are not practical.
Shooting birds in a residential garden, is highly unlikely to be covered by any General Licence.

As for mammals, though they do not fall under the realms of a general licence, in line with the Wildlife and Countryside Act, a shooter must ensure that no undue suffering is effected on the quarry, in the process of shooting. It must be done humanely.
This places quite a big legal burden on a shooter. Using inappropriate kit, or not being able to consistently make a clean kill shot, among other things, could see a shooter breaking the law