If the immersion supply,cable goes directly to the consumer unit then the neutral connection has either come adrift inside or the cable is broken somewhere.
We are now at the situation that you will not be able to isolate all live feed into the consumer unit. In that case I dont think that you will be able to do any safe working and in any case I doubt your competence to do that safely.
My conclusion is that you need to call a competent electrician to open the consumer unit and see why the neutral connection is missing at the spur switch.
You mentioned rewireable fuses earlier.... I don't suppose you have an ancient installation with fused neutrals do you? It's a long shot, but I think there are still a few about.
Did you ever diagnose the old immersion or thermostat as faulty, or did the main fuse blow?
The trouble with a neon screwdriver is that it doesn't tell you the voltage - only that there is some.
So far, I would say the most likely scenario is a loose neutral at the neutral bar in the fuse box.
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