From what I can gather, the "W" on a Singer 319 means it was made in the Wheeler and Wilson factory in Bridgeport Connecticut. It appears that Singer acquired this factory in about 1905. The 319's were made in three places, Kilbowie in Scotland, Connecticut in the USA and assembled in Monza Italy using parts from Kilbowie. Hence the designations 319K, 319W and 319M.
My 319W has a few parts stamped (or painted) as having been made in Great Britain, while the rest are stamped USA.