I don't think it was ever a pub. It was called Old Lamb House (rather than the Old Lamb) when demolished before the formation of Martineau St, and was the home of the drapery business for at least 60 years. In earlier times it was known as The Lamb House, the old being a later addition and there was at one time a malthouse behind it in what was known as Lamb Yard. Close by in Dale End was , in the 18th century, the Beast Market, so maybe lambs were concentrated near the building.
Information from "Faces & Places"