Er, I've got to step in here, because everyone here has it half right.

as far as the law states, you can legally shoot any pest animal, if you've got good reason to do so.

Now, this is where you'll fall into a blackspot, you'll been fine shooting any mammals (Squirrels because of reds, the damage the do to trees, ect)

but, like all Birds in the UK, pidgeons are protected, if you've got pidgeons that are in your garden, and not doing any damage, and posing no real health risk to anyone, it will be illegal to shoot them.

Claiming to shoot a wild animal, pest or not, for food is not legal either. (hence the big debate over pidgeon shooting a few decades back)