Well I heard back from Watson Brothers. My guess had been between 1895 and 1907 based on barrel address (29 Old Bond Street London, where Watsons move to in 1895) and 1907 based on another gun, numbered 29 after mine, being attributed to 1907. Further, the patent for the Webley expired 1902, so presumably before then as mine is still marked "Patent" around the top lever.

Watson's provided a copy of their ledger book and the file name of their image referenced my gun's serial number and the year 1906. So while they've been a bit oblique, I believe they are saying this was made in 1906, or at least sold then. I've asked for direct confirmation of the date, but in the meantime, below is what they shared as text and an image.

What I'm seeing is a 360 2 1/4" double ("A & T" or Arthur and Thomas Watson?) with a lever forearm and standing rear site with leaf, made by Webley. It was a 'Spec price" of 14.0.0. Customer was Profitlich.

In contract, the rifle before mine was a 450/400 3 1/4" case cordite double with Krupp barrels for 17.10.0. The gun after mine was also a 450/300 3 1/4" double, with many difficult to decipher markings for 25.0.0. So I suspect my little rifle was a bargain then as now.

I'm hoping Watson's provides more than just an image of the order book. The only other histories I've ever ordered were for my pair of Dickson double rifles, and those were amazing, with pictures of the ledger entries, but also commentary on the purchaser and context around what volume and mix Dicksons was producing that year.