The OP’s opening post pretty much proved the thread dial that’s now on the lathe doesn’t match the leadscrew that’s now on the lathe. That being the case, that thread dial is useless for cutting threads of any type on that lathe as is.
As I said earlier, setting the gear box for 16 TPI and getting a 15 TPI thread indicates that the gear train was designed and built to drive a leadscrew with 3mm pitch. The lathe probably had a 3mm pitch leadscrew when new. That combo of gears with an 8 TPI leadscrew will give you 15.1 TPI when the gear box is set for 16 TPI. That's why I suggested earlier to set the gear box for a 1.5 mm pitch thread and see if it makes a 16 TPI thread. If it does, that will prove that the gear box was assembled with gear ratios to drive a 3mm pitch leadscrew.
Another way to look at it is, when the gear box is set to cut 1.5 mm threads with a 3mm pitch leadscrew, that's a 2:1 ratio between spindle and leadscrew. When the spindle turns 2 revs, the leadscrew will turn one rev. If the leadscrew has a 3mm pitch, the carriage will advance 1.5 mm for each rev of the spindle, cutting a 1.5 mm pitch thread in the work piece. But, if the leadscrew is changed to an 8 TPI (1/8" pitch), then the carriage will advance 1/16" for each rev of the spindle, cutting an 16 TPI thread in the work piece. I hope the OP gets to run that test soon. Will answer a lot of questions.
Tom