Given Schwinn's high volume, they were able to dictate, to some degree, what Sturmey was producing. Schwinn did use some SW hubs along with AW hubs (I removed a SW from my 1957 Traveler and put on a 1957 AW - Schwinn was using both the AW and SW simultaneously). Schwinn preferred alloy AW hubs when they could get them (not all on the Schwinns were alloy, but a substantial portion were), along with alloy brake levers and brake calipers in the mid and late 1950s (Weinmann 810s usually).
Sturmey had a history of building hubs to order if you would order in enough volume. Sturmey had done the same thing with Sears bikes in the 1910s. The Model S was license-built in the USA during WWI for Sears bikes, particularly the Chief.
Schwinn was notoriously thrifty with what it paid for advertising and parts suppliers. Schwinn used "Schwinn Approved" Steyr copies of AWs for part of the later 1950s, especially in 1958-59. This was certainly cheaper, but the Austrian-made copies were not as good as the British originals. Schwinn did the same thing with Phillips-made brakes in the mid-1950s. There was a brief period of a couple years where Schwinn used English-made Phillips brake levers and calipers. They were subsequently able to get alloy Weinmann 810s at a better price, and so went with those. The Weinmanns were also lighter and somewhat better functioning than the Phillips.