Additionally:
My most recent information, mainly anecdotal, on the export of shotguns from the USA has led me to believe that there is no need for an export licence from the USA providing the shotgun is your own property. This is how I can return unsold guns to the UK after shows.
However, the importer of the gun into the UK must have some kind of import licence (and also a permit to possess the gun) to bring it into the UK.
Furthermore, the gun would be subject to VAT and import duty on arrival unless the importer has an 'Inward Processing Relief' registration.
Lastly, the gun would need an export licence to get it back out to you after the work was done. In this case of a gun made before 1897, that can be done under an 'Open General Export Licence' but the exporter does need to be registered with UK customs to use this facility.
I know this all sound horribly complicated but anybody in the UK who has done this (properly!) even once before will have all the authorities and processes in place and it shouldn't cause YOU any extra work.