A couple of points regarding this.
Whilst its not the best idea to change the reversing switch on a single phase machine capacitor start machine while the lathe is running it won't do any harm because the start winding is switched out (by the centrifugal switch). Any time you change the switch it won't take effect until the motor is started again. N.B, three phase motors are different and reversing the switch will have immediate effect !!!
You can use a simple two pole on/off/on toggle switch as a reversing switch. You wire a pair of crossover wires between the two end pairs of contacts and then feed live and neutral to the middle pair. The feed to the capacitor and start winding goes onto either of the pairs of end contacts. Switching the switch from one end to the other will reverse the motor.
My own lathe does not have centrifugal switch built into the motor, instead it is supposed to use an external current sensing relay to switch out the start winding once the current drawn by the winding allows. Not having this fancy relay (they are still available but they seem to be used/available only in the U.S.A,) Instead I have an extra set of contacts on the back of the start button that power the start winding. Pushing the start button for a second or so holds the start winding energised during start up, once you release the button the start winding drops out leaving the main winding held on via the NVR contactor
Martin