There are few barrel makers plying their trade for the public in the UK, Bill Blacker being the best known. He quoted me about 10000 inc. VAT to re-barrel a Purdey 12b Hgun in steel quite recently.
There are a few extra costs like engraving and blacking etc.
The ease and cost, or lack of, getting the gun to and fro depends a lot on the gun's age. If pre 1897 (UK antique watershed) it is fairly straightforward as the gun can be carried in and out of the US/UK with minimal paperwork.
Once beyond 1896 it gets a lot more complicated and expensive leaving the UK and once beyond 1898 it needs an ATF Form 6 to get it back in to the USA (unless the documented owner is hand carrying it).
There are quite a few issues importing/exporting and someone like Charles at BSA can sort them all out for you but at a price.
