1. Modern immersion heater thermostats have:
1a. An adjustable thermostat, used to set the required water temperature. Once the set temperature is reached (not very accurately) the thermostat cuts off the electricity to the immersion element. The water starts to cool, and after a time the thermostat automatically turns the electricity on again and the cycle repeats.
1b. A non-adjustable thermostat, with a factory set limit, usually around 85 degrees. If the water (or rather, the thermostat) gets hotter than this set temperature, the thermostat cuts off the electricity to the immersion heater. Manual intervention, via the paper clip, is required to reset this thermostat. This thermostat is there as a safety device to stop the water getting hotter and hotter if the normal thermostat fails. Uncontrolled heating can lead to very hot water filling the cold water storage cistern in the loft as it escapes up the vent pipe. Because the CWSC is usually plastic, they can fail if filled with near boiling water.
2. I would suggest as a low cost diagnostic you replace the thermostat.
3. If this does not effect a cure, it would seem the water at the base of the cylinder is continually overheating. As previously stated, this is usually caused by an accumulation of limescale.
4. Unless your water is absolutely soft, 45 years is likely to have caused a good deal of limescale to have been deposited.
5. If you have a blanking plug (look at Screwfix 51855) fitted to the TOP of your cylinder, you could try the following:
5a. Turn off the cold water supply to the base of the cylinder.
5b. Open a downstairs hot tap to run off any hot water in the pipes.
5c. Remove the blanking plug. Expect some residual water, so be ready with old towels etc.
5d. Use a piece of hose pipe to syphon some water from the base of the cylinder. If this has a lot of limescale with it, you will have your answer.
6. Alternatively you could:
6a. Turn off the cold water supply to the base of the cylinder.
6b. Open a downstairs hot tap to run off any hot water in the pipes.
6c. Try undoing the immersion heater. The electricity supply MUST be off. Only slacken it a tiny bit until water starts to appear, then re-tighten.
6d. Drain the hot water cylinder. It should have a drain point at the base, normally where the cold feed enters. However, these drain points rarely work. You might have to syphon the hot water out, via blanking plug (see above) if you have one, or by disconnecting the vent pipe from the centre top of the cylinder.
6e. Remove the immersion heater.
6f. Clean out the base of the cylinder.
6g. Consider replacing the heater and thermostat if you go this far.
7. Warnings:
7a. Electricity off when doing anything with an immersion heater. If you have a double pole switch for the heater use this, but its much better and safer to turn all power off at the consumer unit. Do NOT rely on pulling the fuse for, or switching of the circuit breaker for, the immersion circuit.
7b. Immersion heaters and blanking plugs tend to be incredibly tight, particularly after many years. Always try to slacken them while the cylinder is full of water, as this makes it less likely to collapse.
7c. If you damage the cylinder in any way, you will need a new one.