In 1865 some steel shotgun barrels were produced by a cold drawn process but because of manufacturing and cost problems they were not competitive and their manufacture was discontinued.
Sir Joseph Whitworth patented his fluid compressed steel process in 1874. These dates bracket the time frame in which the first English shotguns were made with steel barrels.
It is interesting to note that adoption of steel barrels was slowed because "Gentlemen" could not differentiate between common steel used in musket barrels and fluid compressed steel! This was likely the reason why Sir Joseph insisted that barrels of his manufacture bear his name and the "Wheatsheaf" trade mark.