Doug,
I know you said you don’t have a dial but do you happen to know the thread pattern on a MacNeale dial spindle?
I’ve tried my ⅜-24 die on the spindle and it binds (as does a ⅜-24 tap on the inside of the drive wheel).
I don’t have a thread gauge but I lined up my tap (upside down) against the spindle and the thread count seems to match. See attached picture:

However, on the spindle, the depth of the threads is shorter, the edges are a bit rounded (worn?), and the angle seems about the same.
Whitworth is 16 tpi while BSF is 20 so I’m thinking proprietary threads - not Whitworth.
Any thoughts are always appreciated.
David