As Peter says - its 'easy' for Registered Firearms Dealers (RFD's) its just a matter of 'willingness' to obtain the export licence and to actually export.
For an 'individual' it is much more difficult and i think the following would be the process :
1) Apply to the Policefor permission to purchase barrel / bolt - wait 6 weeks for the variation to be added to your licence (if permission granted)
2) Buy barrell from Norman Clarke, but as a 'private individual' cannot receive gun parts by either post or courier, Norman Clarke will have to ship to your local dealer (RFD).
3) Your local delaer enters it onto his 'book' (and charges you for the priviledge)
4) You go to your local dealer, and he takes it 'off his book' and puts it onto your licence. The RFD writes to the Police to inform them that he has 'sold' you a barrel, serial number etc etc.
5) You take it home and you write to the Police to inform them you have acquired a barrel - serial number etc etc
Now to sell it 'abroard'
When I sold an Enforcer to CanadaI had to write to the Police and ask for written permission to have it shipped outside of the UK
. This is not an export licence but something additional.
As an individual I didnt seem to be able to get an export licence (or maybe I gave up too easily), so I had to find an RFD who had an export licence, I then had to package up the rifle and take it to my local RFD (and for a charge) he would enter it onto his book and ship it to the exporting RFD, who then could export it.
I then had to notify the Police that I had shipped / sold the rifle to my local RFD, with the final destination / address of the individual buying it.
So pretty complicated.
A UK RFD, such as Norman Clarke can just ship it to any Exporting RFD if they do not want to ship themselves. without all the hassle that an 'individual' has
Edit to add :
Just rembered that in order to be granted permission to buy the barrel (as in #1) you must provide 'justification' and 'good reason'. Stating 'I want to buy it for a friend' would be unlikely to be accepted, if you said it was to replace a worn barrel I believe they would expect to then see evidence of the original being destroyed ('sold' to an RFD)
Maybe a UK RFD who is also an FAC holder could confirm ?