I don't believe the electric immersion element is doing it. Anyway, the cap of yours is scorched which is a bad sign so there may be a fault.
Is the tapwater scalding hot?
yes, it has been - in the evenings when I’ve run the tap for a minute or so it got really hot.
Look at the rectangular silver box with three pipes going into it in a T shape.
Which of the pipes is hot?
None at the moment. When I’ve checked before it’s been the top pipe out of the green tank but that isn’t hot at the moment which would suggest you’re right to say that the immersion isn’t on constantly.
Can you feel or hear the green pump running?
No, tbh I thought that was the pump for the electric shower.
Observe the cream thermostat on the side of the cylinder. Turn it down to the lowest possible sagging. Does it click?
Nope, nothing.
Does the silver box whir and the pump stop?
Negative!
If not, turn the stat down further, then all, the way up. Any click?
No sound from it at all.
(When you have finished, reset it to 60)
When was the silver box replaced, and that horrible cable patch covered in yellow tape done?