Thanks for the suggestions. I don't have a low-value resistor to check the meter with, but I have measured the resistance of our electric kettle (across the plug terminals, as it's one of those 'cordless' kettles), at 32.2 ohms. For 240 volts, that gives a wattage of just under 1800 watts; the kettle is rated at 1850/2000 watts, so good enough agreement there, suggesting the meter is not under-reading.
I've rechecked the immersion heater resistance, disconnecting both ends in case there was anything in parallel, and cleaning the terminals, and get the same resistance at 12.8 ohms. I've just checked the MCB in the Consumer Unit, and it's rated at 16 amps. But with that resistance, the immersion would be drawing 18.75 amps. So why doesn't the MCB trip?
As has been pointed out already, the resistance of nichrome heater wire doesn't change massively with increasing temperature (tables I have indicate 7% increase at 500C. Don't know what temperature the heater inside an immersion element gets to, but it can't be too high with all that water in close proximity). I don't see how a mineral insulated element could 'fail' by going low resistance. An increase in resistance, due to ageing or thinning, perhaps. And if the insulation is deteriorating and there's an earth leakage, I would expect the whole house RCD to trip (100ma, time delayed, overhead mains supply).
I am puzzled!
I can't easily replace either the thermostat or the immersion itself, because some genius 'plumber' has arranged the hot water outlet pipe to go across the top of the tank, straight over the heater and 'stat giving about four inches clearance!
I think I'll explore the option of the Sangamo time switch, as js.B1 suggests.