Some months later, I've finally got around to working out what caused the fire. A poorly designed TIM ammeter backlight bulb holder. A plastic insulator inside it has split, allowing the live centre electrode to short out to the earthed spring inside. This caused the wire supplying the instrument panel lights to heat up to such an extent that the insulation melted and damaged other nearby wires. It even cut clean through the plastic oil pressure gauge pipe! Unfortunately the wire supplying power from the headlight switch to the instrument light switch, even though they physically only about an inch apart, runs through a section of the wiring loom inside the dashboard that has quite a lot of other important wires in it. The headlight circuit on the Morris Minor isn't protected by a fuse; as I understand it the theory at the time was that you wouldn't want a dodgy fuse to cause your headlights to suddenly fail while speeding along a country lane at night.
(Sorry for the rubbish photo)
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The point of my post is to warn others - if you have any add-on instruments with this type of back-light bulb holder in them, check that the plastic insulator inside is intact, and I would strongly recommend replacing them with one of the plastic Lucas bulb holders instead. They are better designed and far less likely to short out:
http://www.holden.co.uk/displayproduct. ... de=070.037
(I've no connection to this supplier other than sharing a surname with them!)
Bad luck that it should cut through the oil pressure pipe as well - could have been very dodgy.