William Moore was an English maker of good quality shotguns. But "William Moore & Co." was a trademark of H. Pieper of Liege, Belgium. Those shotguns are of utilitarian quality but not up to English standards. The use of the name was, as Jim Watson says, an attempt to trade on a well known English name.
FWIW, I strongly recommend against firing any Damascus or twist barrel shotgun. Even though the quality may have been good, and the gun passed proof in 1850 or whenever, over the years powder fouling and corrosive priming salt have been forced into the cracks and crevices. The result is that the barrels often are eaten away inside, even though the outside is nice and the bores look shiny, the metal may be little more than orange lace.
I strongly suggest giving the old timer an honorable retirement.
(Please use a rod to measure the barrels inside and compare with the outside length to be sure the gun is not loaded. A large number of old shotguns are.)