To really be able to do a diagnosis we meed pictures of motor, inside motor terminal block, and inside the starter, but check the following in this order. all with power OFF!
Remove belts and turn the motor, it should spin freely and cary on for a rev or two when given a hand spin.
check for any motor bearing play which could be allowing the rotor to touch the stator.
wrap a couple of turns of cord round the motor pulley and give it a brisk pull, does the motor still spin freely?
Check the motor rating plate and see that it is a 3 phase motor!
Remove the motor terminal block cover and check the connections and the visible condition of the insulation.
Check the smell of the motor, it should smell slightly of varnish and oil, if it has a very strong burnt pungent smell, it is the motor windings that have failed.
The motor will be star connected, three coil ends will be connected together on a spare terminal, and the other three ends will have one phase each connected to them, I am assuming the motor is a similar age to the lathe, and does not have the star connection point buried in the windings,
With a multimeter set to resistance, connect one proble to the star point, and then test from each phase in turn, all readings should be virtually identical within 1-3 ohms of each other is ok.
You really need a megger tester to check the motor windings insulation from earth, but if you get any reading with a multimeter from the star point or any phase to earth (motor case) be suspicious that the motor is at least damp, and needs drying out. If you have or can get someone to megger test the insulation you want at least 1Megohm to earth. Hold the megger test on, or keep winding if it is a wind the handle megger, and if the reading starts to improve, the motor is damp, use a fan heater, hot air gun or even domestic oven on low to dry it out.
Check the belts are not catching anywhere.
Turn the lathe belt pulley, Is the lathe in gear? Is the clutch (if fitted) engaged or disengaged? disengage the clutch, and put the machine out of gear so that the chuck does not turn when the lathe pulley is turned, and then check that the lathe pulley turns freely.
In order to diagnose the starter, we need an internal pic, and also one of the motor plate and inside the motor terminal box. Does the starter "chatter" when the green button is pressed, ie does the amature jump in and out rapidly and repeatedly, if it does you have confirmed that the coil is working (in some way) so the next check would be the holding in contacts, and the overload tripping contacts, but without knowing what type of starter it is I can go no further at the moment, hope this helps!
Phil