I cannot say for certain whether Fordham & Faulkner stopped working in 1920, but I note that in the 1921 and 1925 Sheffield telephone directories there is no listing for Fordham & Faulkner. There is, however, an entry for "Fordham & Fordham, Silversmiths, 17 Cambridge St". Fordham & Faulkner's address was (according to a 1911 trades directory) 17-19 Cambridge St.
A late addition to this topic: Fordham & Faulkner's partnership was dissolved in May 1916 as evidenced by this entry from the London Gazette.
Notice is hereby given, that the Partnership heretofore subsisting between us, the undersigned, William Charles Fordham and Albert Buckley Faulkner, carrying on business as Silversmiths, at 17 and 19, Cambridge-street, Sheffield, under the style or firm of "FORDHAM AND FAULKNER," has been dissolved by mutual consent as from the twenty-fourth day of May, 1916. All debts due and owing to or by the said late firm will be received or paid by the said William Charles Fordham. The said William Charles Fordham will in future carry on business at the above address under the title of Fordham and Fordham. The said Albert Buckley Faulkner will in future carry on business in his own name at 109, Carver-street, Sheffield. -—As witness our hands this 3rd day of July, 1916.