I can’t answer your question but I can perhaps fill in a bit of background until one of the experts come on line.
K4 Yeading Factory was situated a few miles from where I am sitting now. It was a temporary factory in every sense of the word comprising little more than a collection of huts.
I am given to believe that they also used some local engineering companies as feeders including the Rudge Motorcyle factory in Hayes (2 miles away) which was shut down during the war and turned over to war work. Probably they used the toolroom but what else its hard to say. A shutdown engineering company would have been turned over to war work but whether they were producing components for the K4 or not I don’t know. It may however go some way to suggesting an answer to your question. ie an off line production.
Factories were being turned over to all sorts of war production. The prototype and early sten guns were made in a radio factory in Perivale about three miles from there.