No need to worry about gear tooth dimensions, particularly, as long as the DP or Module and Pressure Angle are the same for all the gears involved, in a simple train.
Basically, if you want to cut a thread, you need to compare the pitch required with that of the Leadscrew on the lathe, and set up a gear train with the required ratio.
If the lathe has a 8 tpi Leadscrew, and you want to cut a 16 tpi thread, the Leadscrew needs to rotate at half the speed of the chuck. So you set up say, a 20T Driver, an Idler, and put a 40T gear on the Leadscrew.
For a 12 tpi, you would put a 30T on the Leadscrew.
For 24 tpi you would put a 60T on the Leadscrew.
The idler is only there to “fill the gap” between the Driver and Driven, and to ensure that the Leadscrew moves the saddle TOWARDS the chuck, for a Right Hand Thread.
It does not affect the overall ratio.
If you wanted a Left Hand Thread, you would insert another Idler, so that Chuck and Leadscrew rotate in opposite directions.
For some pitches, you may need to make up a compound Idler to get the required ratio.
Things can get interesting if trying to cut a 19 tpi pipe thread, or a Metric thread on an Imperial machine. (For the latter, ideally a 127T gear gives an exact translation, but a 63T will suffice with a few slight errors, for most short threads
For threads, 3/8″ or 10 mm or smaller, or even 1/2″ or 12 mm, Taps and Dies will make life easier than calculating the gear train for one of the standard pitches.
For instance for a finer feed rate, rather than screwcutting, can be obtained by setting, 20T Driver, 80T / 20T Idler to drive another 80T on the Leadscrew. This gives an overall reduction of 16:1, so that with a 8 tpi Leadscrew, the feed per rev of the chuck is 0.0078″/rev.
If you could introduce another 20T/60T Idler, and use the Tumbler Reverse, the feed rate would decrease to 0.0026″/rev, which should be fine enough for most purposes; especially if the tool is sharp and has a small radius on the nose.
HTH
Howard