EACH immersion heater is likely to have two thermostats (so four in total), one with a dial with numbers on it (the "normal" one) and one which is reset by pressing a button (the "overheat" one). The button may be recessed and require something like an straightened paper clip to access.
If the water overheats for some reason, then one or even both overheat thermostats may trip, as they are temperature dependent. If both trip it would explain the lack of water heating.
If you know how to use a multi-meter there are further tests you can do. However, some of these require the power to be switched on, so you need to be fully aware of what you are doing, and extremely careful:
1. With the power ON - Test that there is 230V between live (should be brown wire) and neutral (should be blue). (Multi-meter on AC voltage range, setting greater than 230V (usually 750V). If not there is a fault external to the immersion heater. Clearly, for the Economy 7 immersion, this can only be done during the powered hours.
2. With the power OFF:
2a. Set the multi-meter to a low resistance range (say max 100ohms). Measure the resistance of the immersion element, which should be around 8 to 12 ohms (cold). The immersion element is normally connected directly to the neutral (blue wire) and to one side of the thermostat, the other side of the thermostat being connected to live (brown wire). If resistance is low, the element has short circuited. If resistance is high, the element is open circuit. In either case the element must be replaced, which requires the cylinder to be drained (at least partially) and is only a DIY job for the very experienced.
2b. Set the multi-meter to its lowest resistance range (continuity check). Connect a lead to each side of the thermostat, then operate the thermostat. At low, the thermostat should be open circuit (infinitely high resistance), at high temperature (or somewhere between low and high) it should indicate very low (.01 or less) ohms, and depending on the meter, give a buzz for continuity. If neither happens, the thermostat must be replaced. This can be done fairly easily, but requires a reasonable knowledge of things electrical.
If you do any work on the immersion heaters be very careful that power is isolated (except for 1. above) and work with extreme care. If you undo any of the cable connections to an immersion heater it is essential that they are done up again properly and tightly, otherwise there can be arcing which can lead to severe overheating and fire.
I'd recommend you restrict yourself to checking, and if necessary re-setting, the overheat thermostats. If you post a picture of the immersion heaters, with the covers removed, it would help in determining if they do have overheat thermostats or not.
Finally, the normal connection is Economy 7 to the lower immersion heater, so that it heats up the whole cylinder at the overnight rate, and the top heater connected to normal rate electricity for a "top up" of heat to the top third of the cylinder only.
