No you cant shoot in London (unless at a licensed shooting club). You can fire an air rifle but not within 50 yards of a road and the pellet cannot exit your property. Finally you are not allowed to shoot game with an air rifle.
You could trap it using a stick, crate, piece of string and some seeds.
What time should I come round on Sunday?
Edit: Given my impending move maybe we can come to some sort of arrangement! Fresh country birds for posh London Pilsner.