Might the German thread actually have been an old metric one, long before these were sort of standardised?
There were German standard metric threads. So I looked it up… No. they start at too large a diameter. The French had a metric thread standard all of their own, too; but by the time of my reference an international metric series already existed*.
BA is actually metric although specified in inch sizes, based on the Swiss, Thury thread; ranging by geometrical rather than arithmetical progression.
So let's have a look at 3 and 7 BA, and 3 and 7 Thury (both by their designation numbers)…..
by O.D (mm), pitch (mm) and TPI (yes, per inch! Probably for lathe-setting purposes.)
3BA.………4.1 0.73 34.8
3 Thury.…4.11 0.79 34.84
7BA.……. 2.5 0.48 52.9
7 Thury…..2.48 0.428 53.13
It looks then as if Bing used Thury threads, which I think arose in the Swiss clock-making trade. Hence the near but not accurate matches of BA to the locomotive's components.
Also, Thury is not a version of B.A.. Instead, B.A. is a version of Thury.
.
*Reference:
Camm F.J. Screw-Thread Manual; Geo.Newnes & Co, London, 2nd Ed., October 1944. pp 139 and 177.