Further to what Bazyle says its important that you do all the thinning on the same side flanks of the threads. Not too tricky if you can do the whole thing in one set up but if working in sections you do need to keep your brain engaged.
A classic "No way will I cock up something that simple!" trap.
Worst thing when re-furbishing a worn screw is that the master, thinnest, thread which all the rest have to be matched to is in the middle. If doing in sections you need a relatively long, nicely fitting bush, to support the hanging end. An ordinary fixed steady isn't enough. I'd be thinking in terms of a solid bush a couple of inches or so long made a close running fit on the screw and held in a solid support bolted to the lathe bed. Although a special support sounds more work than simply setting up in a steady it isn't silly hard to make dead right if you have a between centres boring bar. Done well it gives you the option of working on a short length at a time to minimise deflection.
On a job like this where you want right, really right, a bit more trouble at set-up time is worthwhile effort. Hafta say I've often spent time setting up with general purpose kit to "good enough to do the job if I'm careful" standard when saying "stuff it, I'll make something proper" right at the outset wouldn't have taken much longer and allowed me to "just do it" without crossed fingers and toes.
I've always felt that it should be possible to make a thingy with an internal thread and adjustable cutter that could simply be spun along the thread trimming the unworn parts back to match the worn ones as it goes. Never got my head round how to ensure that all the trimming goes equally on the same side when running both ways from the middle though.
Clive.