I'm not going to start a long online troubleshooting procedure for your P/T, but will suggest a few checks to possibly eliminate the contactor or the motor.
In the photo below, the Mains input are the bottom terminals. Terminal U/2 is the Mains Neutral conductor (blue wire) and Terminal W/6 is the Mains Line conductor (brown wire). You should have a green wire somewhere in that bundle that is the Mains Earth conductor that connects to the green wire to the motor. It could (should) be attached to an earthing terminal somewhere in the box.
The Mains connections are the top terminals. Terminal R/1 is the Mains Neutral conductor (blue wire) and Terminal T/5 is the Mains Line conductor.
The black button in the middle of the contactor is the manual start button, and I usually use this for testing. In normal operation, when the Start button is pressed and the contactor engages, you will see this button pulled down into the contactor.
The white button next to the current overload setting wheel should be the overload reset.
If you are proficient with a multimeter, I recommend verifying you have the correct voltage at the wall-mounted receptacle. If this is good, plug the P/T in and check for the correct voltage between Terminals R/1 and T/5 at the Mains input to the contactor.
If the voltage at the contactor input is good, then connect the leads of the multimeter to Terminals U/2 and W/6 at the bottom of the contactor. You should not see any voltage yet.
While holding the leads on the terminals, press in the Manual Start button in the middle of the contactor. You should see the correct voltage on Terminals U/2 and W/6. If not, then the contactor is bad. If you do see the voltage, and the motor doesn't start and the contactor does not latch, then check for the correct voltage at the output of the overload relay Terminals U/2 and W/6.
If you see the correct voltage at the output of the overload relay, then there are problems with the motor or motor wiring. If you do not see the correct voltage, try resetting the overload relay by pressing the manual reset tab.
Manually pressing the black button in the middle of the contactor removes the start and stop buttons, contactor coil, and auxiliary contacts from the test. If there is no voltage to the motor, then the problem is with this unit. However, I wouldn't buy one until I determined it was the problem.
Good luck!
Update: Thanks to
@Distinterior for pointing out my mistake. I broke one of my three golden rules: Never get involved in a land war in Asia; Never go against a Sicilian when death is on the line; and Never post technical information when I'm tired.