I think the short answer to this confusion is multiple source suppliers with contracts being let to various firms to make the same thing under arrangements more akin to shadow factories than simple contracts used for more mundane items.
The design authority was the Machine Tool Control Rationalisation Board who issued production licences for the standard design to various suppliers. Presumably factory branding was permitted so that, if there were problems, the offending supplier could be identified and to to sort out the issue. I imagine the factories would have embedded ministry inspectors, whether full or part time, to ensure things were kept up to scratch just as the aircraft production folk did.
In the absence of any hard data as to when licences were issued and production actually started its quite creditable that machines from one, or more, suppliers could have been produced in 1939.
Does anyone know when the Machine Tool Control Rationalisation Board was set up? Around 1935 would be my guess as that when the shadow factory ideas were first implemented.
Clive