>>2930221 (OP)(you) can't buy a watchmaker's lathe because none of the suppliers sell to randos. Most require a VAT number or business number before they agree to send you anything. Not that you'd want to because watchmaking lathes are enormously expensive. Least expensive on I know of is the Vector mini, in an incomplete set, for the low low price of 3 800€ via Boley who once again will not sell to (you). I have one in the workshop and if sucks.
Get a modeler's lathe. Look into Proxxon. Sure it doesn't have the ultrasmall chucks for stuff like escape wheel pivots and balance wheen axles but that's not something you're ever going to need anyway.
There is absolutely no reason you would want to get watchmaking gear unless you love paying the "swiss made" Horia tax.
You could go second-hand but every other watchmaker is already looking and will outbid you despite the already high starting prices.
>t. watchmaker