The earliest shotgun that I have seen with steel barrels was a Thomas Horsley hammer gun #1968, circa 1872. This gun was unusual because the barrels were engraved," steel"! I have seen one other early gun by this maker that also had steel barrels.
By 1888 steel barrels were in common use. A debate over the strength of steel vs; damascus, pompted the Birmingham Proof House to carry out comparison tests of steel vs, damascus. The best performing barrels in this test were those made from English machined forged laminated steel in three rods.These barrels out performed Whitworth steel.Details of these extensive test can be found in the book Experts on Guns & Shooting by Teasedale-Buckell.