SKB is right about the UK need for a shotgun or firearm certificate to be in possession of the gun prior to putting it into the airline luggage system. A Visitors SG or FA Permit would do but then the gun details have to be included on the certificate details on application so a impulse purchase wouldn't be possible.
As to the export licence, the rules have changed a bit in recent years. If the gun can be documented to before 1890, it can be exported without any export licence required. If the gun is 1890 plus it needs an export licence which has to be applied for by someone registered with the licencing system. They will ask for an import licence from the destination country (and any countries that it will transit through) but ATF will not do a Form 6 for anything before 1899 so one just politely tells then No, it can't be done. After some checking, the licence is generally issued without the import permission.
If the gun is post 1898, it will need a ATF Form 6 to support the export licence application.
Having said all that, I would not expect the UK Border Force personnel on the front-line to know what the exact regulations are on a gun of specific age and so they would most likely impound it regardless.
Tears before bedtime!
