Well, although there was a Whitworth Standard for Screws for Instrument and Watchmakers, it wasn't included in the British Standard (because BA was adopted for that sort of work.) So can't be BSW in the purist sense.
Probably not Whitworth though because the set is American, and they never went in for Whitworth much! I reckon it's ASME, the US National Standard thread form miniaturised for Instrument and Watchmakers.
Can you measure a flank angle? Whitworth will be 55°, ASME 60°.
Interesting find whatever it is.
Dave