Making DIY Lathe

Do you mean making a lathe from scratch? I really wouldn't bother, as starting with an actual lathe is likely to be easier and give a better end-result.

Lathes are typically easier to convert to CNC than mills, as long as you can find one with space for a ballnut in the cross slide. (though, actually, the nature of lathe operations means that backlash compensation works OK and you can stick with the original screws (up to a point)

Most of the LinuxCNC GUIs have a "lathe" mode.
You might also want to look at:
forum.linuxcnc.org/41-guis/26550-lathe-macros?start=240#247195

I have wondered whether a "gantry" lathe would work well. ie use two ballscrews, one each side of the bed, for Z to hold the X at right angles, rather than relying on a long slide.

I think that it may be too early to worry about the CNC control side of things.
What you need will be determined by the hardware.

For example, with my milling machine retrofit I found a cheap set of 4 servo motors, but they had resolvers, so needed compatible drives and a controller card set that could work with resolvers.

If you are using stepper motors then almost anything can be used. My first conversion used the parallel port and worked fine for a couple of years until I felt the need for more input channels.

One mechanical design tip: Make the Z screw longer than needed to leave space for ball screw covers.
My own lathe retrofit has space inside one of the castings for a spring-type cover:
photos.app.goo.gl/XEpzLSgGRyKf6oDZA
photos.app.goo.gl/2EKJHW6afxtkRjSj6

You might find the whole story worth reading?
bodgesoc.blogspot.com/2015/08/holbrook1.html
(9 sections, index top right)