Kirk Merrington's price for sleeving is $1900 plus tubes (
http://www.merringtongun.com). Of course sleeving a gun in the states will take it out of proof unless you pay another expense for sending back to England. Dig Haddoke estimates 1500-3000 for sleeving in England (
https://www.vintageguns.co.uk/articles/thin-barrels/).
Although the Woodward may have been built by a Birmingham maker, there will be some additional brand value for the Woodward name. An H&H boxlock made by Webley and Scott will bring a premium over the same boxlock being sold by Webley and Scott. I guess people are willing to pay more for the H&H BLE because of some perceived quality control over the standard W&S boxlock.
Rocketman used a BLE for this s example and most people wouldn't spend the money for sleeving a boxlock.
Ken