Condor, I have never noted a barrel set I believed to be Woodward fabricated that was not marked "J.W&S" immediately in front of the barrel flats on each tube. This includes a JUL steel barreled rifle ca. 1879, a steel barreleled top lever hammer gun ca. 1884, and 3 "The Automatic" SLE's through 1898. Excluded would be guns made for JW in Birmingham. Your barrels appear to have a Birmingham mark similar to the one on a JW marked and retailed Webley & Scott Proprietary BLE of ca. 1911 manufacture.
Your gun could have had damascus barrels, but I very much doubt it. 'Ole JW was quick to convert to steel barrels.
DDA