A quick summary of the relation between Douglas and Williamson from details gleaned from Jeff Clew's book "The Best Twin".
Williamson was a separate venture by William Douglas Senior, of the Douglas Brothers Ltd. (soon to be just “Douglas Motors Ltd”.) Many of the components were likely made by Douglas for Williamson, the engine certainly was (and the transmission too it seems.) Douglas used the water cooled opposed twin 8hp engine in their own small car (also a William Douglas venture, but done under the Douglas name), introduced in late 1912. It does not look like Williamson really got going till 1913 or 14, and only had a short run till the War ended production in November of 1916. As noted in the advert they also made sidecars to go with the motorcycle; with a five point attachment no less. They were really meant to be high end sidecar machines right from the start. The short run and high price (few sold) make them very rare today.
After the war Douglas car production moved into the Williamson shops, which suggests Williamson production ended. But if they made a v-twin model up to 1920, then perhaps the venture was sold off and relocated. Not sure where the Williamson shops were, possibly in part of the Douglas Kingswood factory complex. As Douglas were no longer making the Williamson engine, they designed a new engine specifically for the car. Car production ceased in 1922 (rising price and competition from the Austin Seven.)
-Doug