Presumably, the motor ran before the lathe was last used?
Ever the optimist, if the motor is brushed, and has not run for some time, maybe there is a poor contact, brush / commutator, or a stuck brush..
Assuming that everything appears to be OK electrically, spinning the motor by hand may urge it into life, and further running for a few minutes will restore things to as they should be.
If not you need to start tracing just how far the electrical supply is going through the circuit.
I would start at the motor (One method advised is to substitute a filament lamp for the motor and to check if the brightness varies according to the position of the speed control. ) band try to work backwards from there towards the input. Since some of the lights come on, the fuse is intact.Again, taking the easy way out, it could be that the track in the speed potentiometer is dirty.. Possibly, a shot of contact cleaner might get things moving again.
HTH
Howard