CNC style lathe home brew Lathe for drilling and cutting.

[JAZZCNC]

Direct drive Stepper motor won't have the speed or the torque to do this job.!

Really you need a DC motor and speed control from something like tread mill or a servo motor.
Typical Servo motor spins at 3000rpm with linear torque thru the speed range, best steppers will top out at around 1500-2000rpm MAX with very little torque left at this speed.

Collet chuck mounting should be easy enough if you access to lathe, most steppers have a flat portion on the shaft just bore collet shaft and use grub screws and locktite.