The solution may depend on the type of motor...is it a brush'd motor or not?....did you previously see any sparking?...
Such a motor would be highly dangerous in a woodworking environment.It is more likely to be a totally enclosed motor (commonly TEFC).You do not have RCD's...which should be rectified...unless one of those cheapskate and real nuisance installations using one RCD for all the breakers.
That the motor was 'screaming' and not a brush'd motor ...and it should not be one...maybe the bearing/s are faulty and have now rusted. Does your manually turning the durm also freely turn the motor? It is unlikely you have an a/c commutator motor ...unless your machine includes a dc speed controller, in which case it could be faulty. It's more likely you would have a wound rotor than a squirrel cage motor for better starting torque.
You have a very small breaker...6amp on your board...which could be for lights...it would be likely inadequate for your motor start up current...so...which breaker drops out? Were they RCD combos I'd be checking your extension lead connections for a start and then low voltage (250v) meggering the motor.
(Brush'd/commutator) motors can suffer jammed brushes and being stuck to a commutator. Less likely is a short to earth of wire to the brush assembly however it can happen.)
Never mechanically test the machine with the power applied even with switch 'off'...disconnect entirely.
On the look of it , and the breakers as an electrician arriving there I'd quickly check lead and voltage even though the fault shows only when the machine is turned-on... suggesting the fault is in the machine. I'd be checking a long shot first...that the extension lead has the correct polarities, same at both ends and matching the appliance plug.
I'd check your work to see that in trying to remove one fault you have not introduced another. I see you say the lead operates other appliances including a lathe so it seems the extension lead may be ok. Presumably...??.. the same extension or one like it was used from the same outlet when it was previously working.
As you have apparently no safety switches...although somewhere I saw you don't know which breaker it is...if that's the case is it your main switch, which could be a safety switch, 'dropping out' . If so the problem is likely earth leakage or a faulty capacitor start if cap star/cap start-run. If the motor is started with a centrifugal switch (did you used to hear 'click' just after it started) it could now be a be 'welded' or spring broken so the start winding is permanently engaged.) Typically you would hear a hum owing to current drawn when the motor cannot switch into 'run' winding...then breaker out...but that's not god speaking.
Check the connections at the motor...remove the junction box cover...just see every thing is tight.
You are not an electrician so don't get too involved. Don't engage in 'the blind leading the blind' repairs. Don't mess-about!
You could check the machine switch if it has one (hopefully industrial with an easily reached 'off' switch mounted in a box ). to just see if it looks ok inside ...nothing broken or loose nothing obviously shorted but go no further.
Sensibly guineafowl suggested rodent damage but if the motor is TEFC that will not happen inside the motor itself.
As I said it's unlikely if the tool is of good quality and was compus of safety in design, to be spark producing...I can't answer for the Chinese rubbish sold today to DIY suckers of course. Try your tool in a different circuit GPO if possible. What happens now?...if c/b interrupts you have isolated the problem to the machine itself.
By the way it is not 'pathetic' that leads on such appliances are short...the idea is to minimise leads lying about but in DIY situations it's unlikely to find much common sense in safety but more common to find people who mock safety, effectively beating their chests in a macho fantasy but closing down brain, as other discussions have shown. We see that regularly at the whitehouse......
You should have an outlet very close to the machine. If not, have one installed by a licensed sparkie. I'd be inclined to put a circuit board in your shed with an RCD combo 20 amps feeding your outlets. In the event of c/b operating or you being hooked up no one has to run to the house to isolate the circuit.It's also more likely you'll be comfortable with your installation.
You should not have extension leads lying around. They get damaged and some faults can electrolux you...for example 'crushing' breaking the earth wire and shorting the active to it on the machine side, not the breaker side.... It happens
Voltage drop along a long extension lead can produce a fault in motor starting but you say it enables other power tools to function probably 1/3rd horsepower on the lathe.
If the thicknesser drum is belt driven, as a final test you could disconnect from power, remove belt/s and see what happens when power connected. If still dropping circuit breaker...and try not to do it over and over as the breaker could become damaged...then you will have pretty well isolated the problem to the switch>>>lead to motor>>>motor as the problem area.
At that point I'd be 250v (not 500 or 1000v) 'meggering' the plug between active and earth and neutral and earth (in case you do have an RCD) and active and neutral firstly switched-off then switched-on (no power applied or connected ) Depending on the outcome I may then pull out the motor.
Take it to where ? you asked...well there may be a motor re-winder nearby who can check and quote on repair or its likely you can buy a new motor somewhere locally...... the Italians make many and export many and have pretty good prices....for less than repair costs. Of course it has to match specs of the original one., but should be TEFC (totally enclosed fan cooled) or if a dc motors, also enclosed... By the way the worst battery drills I have found for sparking are deWalt...I wouldn't use one in a saw-dusty area.
Motor bases are standardised now. You'd have no trouble matching your motor to a new one. You will have to match shaft size, physical size, speed (number of poles) and ensure your pully/pullies will fit properly...Voila.
