Nial,
Each Police Force in Britain seems to interpret Firearms Law in a different way. Best to get your son to contact the Firearms Dept at the County Police HQ. If your son is a shooter and has a secure place (safe or gun cabinet) to safely store the shotgun when it is not in use, then there is likely to be little objection.
Temporary Firearms Certificates are available for rifle shooters visiting the UK so a shotgun permit (temporary) shouldn't be a great hardship. Note that Pump actions and semi-automatic shotguns are prohibited as are shotguns with barrels less than 24" in length.
Get on it now though, don't leave it until just before you arrive. Just in case there are questions you have to answer to satisfy the authorities. I don't know if there are any charges, but for a temporary permit they shouldn't be much at all.
Sadly, you are going to be at the wrong end of the country for top class Wood Pigeon shooting. Northumberland is the place to go but it's 300 miles away from Surrey. It's a great sport and the meat (breasts)make very good eating.
Have fun.
Harry
Last edited by Harry Eales; 03/04/10 05:46 PM.