[Station X]

If the motor is making a sound, that means that power must be getting to it.

Either the motor is faulty or it's so stiff that it cannot rotate. If the latter, then lubricating the motor's bearings with say clock oil may cure it.

Any chance you could measure the voltage on the wires going to the motor? I assume they terminate on the Printed Circuit Board (PCB) somewhere?

Is there anything else on the PCB apart from the transformer and two resistors?

EDIT. I've just had a closer look, and the yellow gear isn't actually on the motor shaft. It appears that the white gear is mounted on the motor shaft and drives the yellow gear. Is it possible that the motor shaft is in fact rotating, but the white gear is slipping on the shaft due to say a cracked hub?

You'll have to work out the gearing arrangement. The fast running motor will be geared down so that the units half of the minutes display rotates once every 10 minutes. That means that small gears will be driving large ones.