Thanks for the suggestions. I've had a go at testing continuity at least along the control module pictured (see attachment with labelled terminals). I've also tried the element and I'm getting wonky results which I don't understand.... I will explain below.
By the way, the control module is completely disconnected when I am measuring it. It is not in circuit. Same goes for the heater element (which is still screwed into the base of the tank).
Here are the results:
1. From the picture I posted, I get continuity along the left side from #1 --> #2 ---> #5 ---> #6.
#1 is the black input from the mains. It is continuous to #2 which indicates the RESET switch is working ok and not tripped. There is a metal contact physically connecting #2 to #5 so obviously they are really the same node and continuous. #5 is input to the thermostat control/limit, and #6 is the output. I have continuity between #5 and #6 as well, indicating the thermostat limiter is closed switch (since tank is cold) and has not failed either. So #1 --> #6 is all working.... and #6 connects up to the heater element.
2. From the picture posted, I get continuity along the right side from #3 --> #4.
#3 is the white input from the mains. It is continuous to #4 which also confirms the RESET switch is working. If the RESET was tripped, it wouldn't connect #1 to #2, and #3 to #4. Then #4 is connected to the other side of the heater element.
So far... all ok. The mains BLACK at #1 is getting through to #6 and to one side of the heater element, and the mains WHITE at #3 is getting through to #4 which is connected to the other side of the heater element. There must be electricity getting to the heater. I could connect a light bulb socket to #6 and #4 and it should light up. If I use say a 30W bulb, at 120 V, that will put 0.25 A across the RESET and thermostat (P=VA=120V x 0.25A=30W). I know the cold bulb will draw higher current, but it shouldn't trip the RESET and I would imagine quickly settle to 30W and give me a glow indicating I am getting power from the mains to the tank and through the control box. The only variable then would be the heater element.
Do you think that light bulb test is a good one to run next? I don't want to touch live mains. I can hook it all up and stand by the fuse box, trip the mains fuse and see if my bulb lights up from a safe distance away.
3. The heater element/coil - I'm a bit confused.
I hooked up my DMM on ohms across the terminals of the element. At first it was showing resistance in the few K, then started to rise, then it went high to many M, then it started falling again, never quite settling. As soon as I take off the DMM and try it again, it starts at a different number. It never gets to some stable number. At first I thought maybe there is some "inductive" effect happening or even capacitive, but I can't rationalize it because if it is just a simple element/coil it should reach a value fairly quickly.
Anyways, for now the heater element seems to be giving me very strange readings measuring ohms on my DMM. So I'm a bit confused. I am more confident that the RESET and thermostat limit part is working correctly, and that the coil is buggered up but that still doesn't explain how the ohms changes all over the place and creeps up and down seemingly randomly. I must be doing something wrong.
If the coil was broken, it would be infinite resistance, open circuit, not vary with time?