It's certainly not 5BA, whose o.d. is very close to 1/8 inch.
(8BA is actually 0.126 " full diameter, and a lot of model designs specify 1/8 " dia rod for 5BA studs, so within tolerance.).
I still think 7BA but I'm surprised an 8BA nut will fit.
SO
7BA 0.098 " full dia, pitch 0.019" : 2.5 mm 0.48mm respectively.
8BA 0.087 full dia, pitch 0.017 " : 2.2 mm 0.43mm " "
Diameter measured: 0.085" (2.16mm)
Pitch from the photograph, very slightly <0.5mm
Inch values rounded to 3 dec. places from 4-figure tables. The sizes look odd because the full BA range is a metric geometrical series.
If it's none of what anyone's suggested it can only have been some non-standard thread of the maker's own.
Does anyone have a spare 7BA nut they can give, or a 7BA tap they can lend, to Nearly Done? (I might have a tap, but not a nut.)