Have you also watched the whole Clough42 YouTube series? From memory, his lathe is roughly the same size as a M300 so I would use that as a guide. I would guess something like a 3Nm NEMA23 would be about right and run off 60-70V to get max torque. This is a lathe leadscrew, not a ballscrew driving a router, so it's torque rather than speed that's needed. So gear down ("belt down" doesn't sound right but I would use a toothed belt) from motor to leadscrew. While the leadscrew is pretty chunky so a fair bit of rotational inertia, the saddle is a heavy old lump as well with relatively high friction sliding surfaces and you have to get that moving.

For this job, I would go stepper not servo. Closed-loop stepper, maybe, as they are not that much dearer than a plain stepper these days, but wouldn't go servo. Costs more, you don't really need the extra torque or resolution, and in fact the higher resolution of the servo might be a problem as you need a higher pulse rate out of the controller module which might push it a bit. Not done the sums to check that, though.

I keep thinking about doing the same for my lathe but it would need a bit of modification to get the drive to the leadscrew and I haven't worked up the energy yet - or needed to prioritise this over the rest of the project list! - to figure out the details. Nice project, though.