Actually, it was a bit different....
The swap among the GM lines was to keep production going; by this point the majority of all GM cars built were ordered with H-D by the dealers selling them. As for the independents, Willys, who just started using H-D in the Ace and Eagle lines had to go without for a number of weeks. Kaiser had enough Hydra-Matics in stock to cover its needs until production resumed at Willow Run. Nash, Hudson and others also had to suffer until the end of 1953 when Willow Run-built HD's got into the pipeline in volume
Lincoln used H-D into 1955 models if memory is correct. It was with the 1956 line up that the in-house automatics at Ford got into the top-line cars.
I have a newsreel clip showing GM setting up the Hydra-Matic line at Willow Run, including taking large machines in through the big hangar doors, with the Kaiser-Frazer name still showing above them. I got that back when I was a historical consultant to GM, Ford, the University of Michigan and other parties over the pollution of Willow Creek. Barbara and I went do to meetings and stuff in Chicago on weekends and we were put up in the Palmer House or the Union League Club.