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The only book covering Sheffield exclusively of which I am aware is "The Sheffield Assay Office Register" by B.W.Watson, published by Wm Townsend & Sons, Sheffield, 1911. This has pretty good coverage - although obviously not beyond 1911. Unfortunately it doesn't show your W&Co mark.
This is the mark of William Wild & Co who registered with the Sheffield Assay Office on the 23rd December 1784. The mark registered normally has two W.W & Co but often you see it as W&Co as with your example. The company were based on Trinity Street, Sheffield.