You could replace the motor with one that can vary in speed. If you kept the stepped pulleys it would allow speed ranges.
A DC motor would need a DC power supply and for it to hold speed well, a sensor/tachometer would need to measure the shaft speed of the motor and feed this signal back to the special power supply to increase the motor current as the load increased, and reduce it as the load decreased.
An AC induction motor varies it's speed when the frequency of the AC power going to it changes in frequency. There are single to 3 phase controllers that can now be used to provide 3 phase power to a 3 phase type induction motor, and these will allow you to change the speed of this kind of motor. Don't try to slow a single phase induction motor by reducing it's voltage. It will loose power as well as speed, and when slow enough, the centrifugal start switch will close and energize the motor start winding, which is not designed to be energized for more than a few seconds at a time, and this winding will burn up rather quickly. A three phase motor has no centrifugal switch or start winding, so does not have this problem at slower speeds, but can have the cooling problem due to reduced fan speed.
I caution that with either AC or DC motors, the motor fan is usually connected to the motor shaft, so slowing the speed of the motor significantly when under load will result in the motor overheating. Keeping the speed at 50% or above will usually prevent overheating. Having the step pulleys between the motor and load will allow you to run the lathe very slow and still be variable in speed within the slow pulley/belt position without slowing the motor to less than 50%. Large motors that need a wide speed range are built with an auxiliary fan that runs at a constant full speed that is mechanically attached to the larger motor and it drives a fan to provide full cooling of the larger motor at any operating speed or load.
Charley (an EE in a former life)