I was fixing those today. The transformer is 56VAC open circuit, but drops to 2V with the element. the timer is really simple, a half wave rectifier, a 4u7 capacitor and then a power resistor to drop 300V to drive the relay with 24V. The timing is set with a RC delay, with the 220k pot on the front providing the dalay along with a 47k resistor ( minimum time setting value) and a capacitor ( 10uF or 22uF depending on what they found), with a 6V8 zener diode driving the base of a NPN transistor that then drives the gate of a SCR ( CR02M on the originals, but I replace them with C106D when needed) that fires to short out the relay coil. LED is in series with the relay coil.
Most common failures are the relay burning a contact, and shorting to the coil cooking pretty much everything. I also add a VDR to tame the spike across the contacts as they open. Mechanically the elements burn the ends, and then you need new end pieces. The switches also burn out, but are easy to replace. I have enough of these at work, and keep a lot of spare parts for them.