Note that Pump actions and semi-automatic shotguns are prohibited
AFAIK, both are not banned.Only the licensing procedure is different.Upto three shots,both can be on a regular Shotgun Certificate.More than three rounds capacity requires a Firearms Certificate which many do have.
Pigeon shooting is great fun and it is not hard to get.Plenty people offer a day's wood pigeon shooting for about $55-60. You can even borrow the estate's gun.You do not need to have a shotgun certificate provided one of the estate guys is with you.Not difficult and not expensive.I think there are a few opportunities right there in Surrey.
Best-
Vikram