Summarizing what I have see here - the motor starts OK, but stalls as soon as load is applied. The lathe in question is a fixed speed lathe using belts and pulleys to change speed (ie, not electronic speed control and no reeves drive).
The symptom is that the lathe cannot deliver torque to the workpiece. That means that one of three things:
1 The motor is unable to develop full running torque
2. There is 'torque leakage' in the mechanical linkage between the motor and the spindle.
3. There is excessive friction in the mechanical system that is increasing the load on the motor to the point where incremental loading on the spindle stalls the motor.
Everett has checked the tension on the belts, and has also confirmed that the pulleys aren't slipping. That would appear to rule out torque leakage. And the absence of noise in the bearings would rule out excessive friction. The third option is that something has changed to reduce the amount of torque that the motor can product.
What kind of motor is this? If it is a 'capacitor run' ac motor, the problem could be that the capacitor is failing. Another possibility is that there is a bad connection to the stator winding that is causing excessive voltage drop and reducing the torque that the motor can produce. I would not expect to see a dc motor absent some kind of electronic speed control, but if it is dc, then the problem could be worn brushes.