Gun
Providing you have a valid shotgun certificate / firearms certificate, you should be able to transport it as if it were secured in the boot of your car.
You would need to have your license with you to make the purchase and the gun's details would be recorded on the license as normal. The dealer would have to complete a transfer document and you a receipt document which then get's faxed to the police authority (not sure if it is just one in the UK or regional).
If in doubt contact your local police authority and ask, but if it were me and I was travelling in the UK, then I would do it.
Taking it overseas is a different matter - Not all carriers have locked storage for firearms.