I've got a lathe. An old one.
It's a Stark jeweler's lathe, uses #3 (sometimes called 3SS) collets, and it was old when granddad bought it. Probably made 1890 +/- a decade or so.
It's still mainly usable, but examining some collets I see the threads are deformed, and I suspect the drawbar has worn excessively (so only the crests of the threads are under stress).
It's time to make a new drawbar, but my problem is this: the old references say the threaded part of those collets (actually called split chucks in the catalog) is 0.520" 26tpi, and they seem to measure at 13mm 1.0mm pitch, which is 0.512" 25.4 tpi.
So, I could buy a M13 x 1.0 tap, and reform the collets and build a new drawbar to fit, OR I could get a custom-built 0.520" 26 tpi tap and keep it historically accurate (the collets might be worn and it'll be a bit loose regardless). One of these is more historically correct, but WHICH?
In that era, it's likely the collets were made in Europe and with metric tooling, Stark being the US agent. My references might be from aftermarket measurements, and I'm unsure how accurate those numbers are.