As, in the UK, a barrel is a 'pressure bearing part' it requires the same licencing as a complete rifle, so I would guess (that's all it is) that the export procedure would be the same as for a rifle.
I have exported Lee Enfields to the US and Canadaand basically what I needed to do was to get permission from the Police to sell / ship the rifle out of the UK, and tell them
1) the name of the final recipient,
2) the name and details of the importing company
3) and the name of the UK export shipper.
I ship the rifle to the export shipping company (at Heathrow Airport) they apply to the UK Government for an export licence, which is granted once the receiving import company confirm they have an import licence from their government and it is then put on an aircraft & shipped.
The final recipent then pays his importer freight, taxes, and fees and collects it from the importer.
I tell the UK Police it has been shipped (document from the export shipping company) and it is then 'written off' from 'existing in the UK'