I have no problem with you shooting them, in act if local I’d do it for you with a clear consciences.
Couple of points to take into consideration though.
Depending on the size your garden, if you’re using an air riffle don’t let any pellets stray across your boundary, classed as armed trespass or something daft!
With a legal limit air rifle you’ll pretty much have to shoot them in the head as unless you’re very close you wont get through the breast feathers with a chest shot.
Easier to bait them down and shoot them on the ground. If shooting up at them into a tree you have to aim lower than you think to compensate the elevated trajectory. Not as easy as it sounds when you trying to hit something the size of a pigeons bonce.
If you can easiest way to blast the blighters is with a shot gun but they tend to upset the neighbours.
Finally make sure you can tell your woodies from your ferals.
As mentioned woodies are a tasty bonus or why not try your hand at taxidermy (?) but treat ferals as you would rats, they can carry just as many nasty’s so pick up with gloves and burry them or burn them.