Like Chris said, Cleaver firearms at Margate would be the best around, he will have everything you are after there.
I don't shoot anymore but when i used to own guns for pigs i used a 6.5 x 55 M96 Swedish mauser, at the time i chose this calibre rifle for its power, accuracy and cheapness of ammo, and it always did the job well. You can't go past the good old 303 sport rifle either.
For bunnies i used a 12g pump action but as you know we can't have them any more, so now days i'd go with an under and over. Another rifle i owned that was great to use on the bigger animals was a stainless Ruger mini 14, i think the calibre was either .223 or .243 i can't remember, i also owned it's smaller brother the Ruger 10 22 carbine, it was fun to use on foxes rabbits and great for target practice. Unfortunately you can't own these guns any more as they are semi auto.