well...I wish it was that clear cut.

The problem could be as you say....

or it could be that because the timer was knackered, the heater was
just always on (boiling the water to buggery), thus never had a
problem.

or it could be in addition to this that the immersion heater is utterly
calcified, perhaps in particular in the bit that contacts the cold
water first....and hence needed to be on forever to keep the water
hot.....

I just don't know.

The fact I can't replace the apparently original immersion heater means
I can't check it, and also means if it does die in the future, I'm
still going to be in the same situation as I am now.

The tank isn't in great condition, and as for being water
tight.....well.....think it probably was at some stage, then
wasn't....got repaired and now that repair is getting a bit iffy - as
feeling round the immersion heater nut last night when trying to undo
it, I noticed my fingertips were getting wet.