Measuring the screws and relating to the BA thread system got me confused, so I had to learn more about it.
Its a British standard, so I expected the pitch to be imperial - but none of my imperial thread gauges matched the pitch of the early Supersport/Superstar rear screw. (clearly not a 30TPI, almost a 32TPI). The closest thread gauge I found is a 0.8 mm pitch - on a British thread system???
Time for some googling!
This page cleared up my confusion:
https://journeymans-workshop.uk/baandme.php
So, the table lists the 2BA as .185" diameter x 31.4" TPI!!! That does not really sound logical..
But the text goes on to explain:
"British Association threads were a metric thread system devised for small screws. Originally included even smaller sizes intended for use in watchmaking but these have been superceded by specialist horological thread series. Not metric like you might expect, but with diameters determined by a factor proportional to a power of the logarithm to the base 10 of the thread pitch in millimetres. Hence the rather strange mix of metric and imperial in the above table."
Aha! Dividing 25.4mm (1") by 31.4 TPI gets you 0.809, so the close to 0.8mm pitch starts to make sense.
The 4.6mm I measured with worn calipers is close to the listed .185", which translates to 4.7mm.
Its a confusing (when compared to more recent thread standards) system, but the 2BA really is close to a M4.7x0.8 thread.
Which is good for re-tapping to a M5x0.8 thread!