https://thevinylfactory.com/features...r-own-records/

The basic process predates NC, but I'm not sure if it's a process that could realistically be done using NC, given the frequencies involved.
Moving the recording head could be done very easily using NC, but the actual cutting at any kind of productive speed, I'd doubt. It's the kind of process where an analogue process (i.e. a electro-mechanical system) will greatly out perform a fully digital process.

I'd be searching to try and find out how the recording/cutting heads on the old school disc recorder heads work, like those in the link I found above. If you can create a working head, then the actual lathe part should be pretty straightforward.