I find it a little strange that you have an English lead screw and threading name plate, but have gears that fit a metric shaft, 18mm? Similar to what @DAT510 showed. Maybe the lathe was designed for metric and then they just changed the lead screw and the gear teeth numbers to go to English. You measured 18mm, 10mm, 23mm: i.e. 18/25.4(*64ths)=.70866(*64ths)=45.35/64. If it were really an English size I would have expected the number of 64ths to be closer to an integer. If you have the gears off also measure the mounting shaft(s) just to be sure as it appears that you may have been measuring one of the gears that you had it sitting on top of the lathe in your first pictures.
Also, why not set the lathe up like the manual might indicate, set the gear box to a thread that easy to measure and let the machine make a light cut (scratch) so that you can see if it really makes the TPI you might expect. For example, 10TPI, 24TPI, or others are easy to measure or match to a known bolt. See pages 9-10 of your manual, and maybe page 13. Figure 10 shows a picture of your gear positions and at the left hand side of the table shows the gear set up that is suppose to be for that table. I cannot make out the bottom gear T number, but I looks like the top gear is a 40, the middle gear is 127, and the bottom gear is either another 40 or 48. Anyway, ratios would be 40/127, 127/127, and 127/46 or 40. The middle gear only transfers the motion since the same side is use for both of the others. So the over all ratio would be (40/127)*(127/127)*(127/(46 or 40))=40/(40 or 46). I would guess that it is the 40. So the ratio is simply 1! The manual also says that you have a set of English change gears (page 13) of 30, 32, 46 in the tool box in addition to the two 40's that are mounted on the lathe. These numbers agree with the numbers that @DAT510 posted. So on page 13 it also says you would need the following for the metric version of the lathe which would have the 3mm pitch lead screw. These would not work for you lathe since the lead screw is 8TPI.
Anyway, from all of the information that you have you can compute a set of tables of gear box gear ratios for every gear box setting. (spread sheet). Then from this you can figure out the TPI for each of the combinations of change gears that you have more might want.
You are lucky. You have a manual for your lathe! Although, some of the sentences read like they were written in China, not Canada! For example it refers to the "lead screw" as "screw leading stick"? Anyway, the tables look informative.
Have fun.
Also, why not set the lathe up like the manual might indicate, set the gear box to a thread that easy to measure and let the machine make a light cut (scratch) so that you can see if it really makes the TPI you might expect. For example, 10TPI, 24TPI, or others are easy to measure or match to a known bolt. See pages 9-10 of your manual, and maybe page 13. Figure 10 shows a picture of your gear positions and at the left hand side of the table shows the gear set up that is suppose to be for that table. I cannot make out the bottom gear T number, but I looks like the top gear is a 40, the middle gear is 127, and the bottom gear is either another 40 or 48. Anyway, ratios would be 40/127, 127/127, and 127/46 or 40. The middle gear only transfers the motion since the same side is use for both of the others. So the over all ratio would be (40/127)*(127/127)*(127/(46 or 40))=40/(40 or 46). I would guess that it is the 40. So the ratio is simply 1! The manual also says that you have a set of English change gears (page 13) of 30, 32, 46 in the tool box in addition to the two 40's that are mounted on the lathe. These numbers agree with the numbers that @DAT510 posted. So on page 13 it also says you would need the following for the metric version of the lathe which would have the 3mm pitch lead screw. These would not work for you lathe since the lead screw is 8TPI.
Anyway, from all of the information that you have you can compute a set of tables of gear box gear ratios for every gear box setting. (spread sheet). Then from this you can figure out the TPI for each of the combinations of change gears that you have more might want.
You are lucky. You have a manual for your lathe! Although, some of the sentences read like they were written in China, not Canada! For example it refers to the "lead screw" as "screw leading stick"? Anyway, the tables look informative.
Have fun.