Hmm.
It is better to use a leadscrew and change-wheels (or screw-cutting gearbox – which amounts to the same thing though is less fiddly to use).
If you can, use a much coarser pitch than 1mm. Without knowing the overall size of your lathe, I would suggest 3mm pitch as the mimium – this is pretty well the metric equivalent of the 1/8″ pitch on a 3/4″ dia screw, used on many small lathes such as the Myford series designed primarily for inch-based work.
It’s best to use a trapezoidal form of lead-screw thread, and this, together with appropriate nuts, is available as threaded stock rod from the likes of HPC Gears and other transmission-components stockists. These are of ground finish, intnded indeed for making lead-screws.
I don’t know trapezoidal metric screw-thread forms but would think from the ISO-M series of “ordinary” threads, a 3mm-pitch one would have an outside diameter about 20-25mm.
You could use standard studding, which is of right-hand thread (an important point I’ll come to) but it and the nuts will probably wear quite rapidly. If you do, then go for stainless-steel studding. From experience this has a decent surface finish whereas plain old zinc-plated mild-steel studding is often rough old stuff for holding the workshop together, not precision machine components! Use leaded-bronze or cast-iron (or aluminium alloy??) for the half-nuts on this.
Note the hand of the thread. It is commonly left-hand but can be a right-hand thread, with due regard to the drive from the spindle. That is normally via a tumbler reverse gear anyway, so you can cut threads of either hand.
You would need fit half-nuts, aka clasp nuts, otherwise you have to wind everything backwards for each iteration of the screw-cutting.
A lathe set up like this also needs a “Thread Dial Indicator” to show you when to re-engage the nuts for the next bite, so the tool always engages the thread correctly. It consists of a pinion, normally but perhaps not compulsorily of worm-wheel form, of appropriate pitch engaging the lead-screw, and directly rotating a dial with, usually, 8 divisions (at least on inch-pitch lathes). It is designed to be swung out of engagement when not in use, to save wear on this relatively fragile component. It allows the lathe to continue revolving, and in the same direction, throughout the operation, saving time but more importantly in our context, wear-and-tear on the motor and machinery.
The clasp-nut assembly does need careful fitting to ensure smooth operation, so each turn of its own thread always lines up when you close the nut around the leadscrew just when the TDI tells you.
Get it all right and it makes single-point screw-cutting a pleasure!
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Jason suggests a groove (properly, a key-way) along the leadscrew to drive the normal turning feeds. This transfers the motion via a pinion bored to a close sliding fit on the leadscrew, with an internal key to engage the groove.
You can do that, I think some small lathes are so fitted; but if possible a separate, parallel, plain shaft with just a keyway for that function, is the better option. The headstock has a [screw-cutting – neutral – feed] selector mechanism on the gears linking the two shafts to the change-wheels so you can’t accidentally engage both at once.
Since you are fitting power cross as well as long feeds it’s wise to fit the feed direction selector with a simple locking control to prevent inadvertently moving it straight across its central neutral point and engaging the other direction. This can be interlocked with the feed ON/OFF lever.
(It was horribly easy to make that mistake on my old, IXL-badged Erhlich lathe. I used a bolt in the control’s locking-pin hole as a safety “blind”!)
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A little extra since you have the luxury of designing your lathe from scratch:
If you can fit a trip-gear to the half-nut control, that is a bonus. It is adjusted to stop the travel when the tool reaches the end of the thread, and until you wind the cross-slide back the tool simply produces a groove round the work at that point. This is normally harmless. I usually cut a run-out groove to depth with a thin parting-tool anyway, if the work allows.
(Production-lathes intended for regular screw-cutting have more elaborate arrangements that stop the feed and retract the tool. A clue to that having been used is the thread stopping at a little run-out ramp instead of a radial groove right round the component. )