Measuring a (maybe worn) ACME

Cutting acme threads is no more difficult than cutting 60 degree threads. Once you get past the chatter you'll make it work. If it were me, I would cut the screw to fit the worn nut.
 
A very mundane question, I know, but how does one discover where the wear is in an ACME thread rod and nut?

No question, the pair are a bit in trouble! The rod is the 1/2" x 10TPI left-hand ACME used on a tailstock.
There's quick-and-dirty solutions, like this one
<https://www.practicalmachinist.com/vb/south-bend-lathes/quickie-repair-worn-crossfeed-nuts-273712/>
but maybe there's a second cause: the thrust bearing in the tailstock could be the loose element,
it'd have the same play effect as if the threads were worn.
 
Cutting acme threads is no more difficult than cutting 60 degree threads. Once you get past the chatter you'll make it work. If it were me, I would cut the screw to fit the worn nut.
Thanks Chuck. Now that is an option I had not considered - and why not? This time, given that the whole quill was wrecked by earlier misadventure before I got the lathe, and has to be re-made anyway, a new nut is to be part of that. I am using one reasonably condition South Bend 9C lathe to fix up parts on the less fortunate SB-9A.

I got some PB104 phosphor bronze, and I am happy to attempt making a new nut insert for it.
So can your suggestion be varied to turn it around, and cut the nut insert with slightly thicker threads to sweetly fit the screw instead?
 
There's quick-and-dirty solutions, like this one
<https://www.practicalmachinist.com/vb/south-bend-lathes/quickie-repair-worn-crossfeed-nuts-273712/>
but maybe there's a second cause: the thrust bearing in the tailstock could be the loose element,
it'd have the same play effect as if the threads were worn.
Wow! This one is quicker and dirtier than most! Perhaps not-so-dirty, given how "clean" things need to be to make it work. I will keep it in mind. This time, I am going to at least try for the "make it from scratch" solution as a starter.

Making a zero-backlash rework like that one has something in common with the heat-and-form-to-shape trick using a Delrin nylon nut, but using tin on the screw instead.

You are right about the other cause of backlash. Taking up the sloppiness at he thrust ring is as simple as using the right spacer. For this case, I was waggling the screw in the nut with all disassembled. The poor fit is real! Back end thrust disc sloppiness would only add to it!
 
So can your suggestion be varied to turn it around, and cut the nut insert with slightly thicker threads to sweetly fit the screw instead?
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I would much rather make the screw than the nut. It's difficult to cut small diameter internal acme threads.
 
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