The Sturmey Model S is a unique chapter. Around the time that the First World War started, Sturmey contracted with Sears to produce tri-coaster hubs in the USA. Sears supposedly paid a base fee of 500 pounds for the design, and then paid a royalty for each hub produced. Sturmey got the money and its products continued to be manufactured and sold during the first few years of the war, while production of war goods took precedent in Britain. It offered something for both sides.
The shifter generally seen with the hub is the flat face quadrant with diagonal "Sturmey Archer Patent" on it. It differs in size and construction somewhat from the 1930s era "dome face" quadrants more usually seen in the USA. It also has different wording than the subsequent "Model K" inscribed quadrants. It is conceivable that the old-style "barrel" handlebar shifter could also be found with the hubs, but the quadrant is what you normally see.
Both the hub and the shifter are difficult to find today in complete and good shape.
Shortly after the First World War ended, Sturmey brought out the Model K series of hubs, which included the Model KC of the early 1920s - a tricoaster based on the K that was to replace the Model S and Model F.