Popping in to A4Audi again this afternoon to get this and the parking sensors looked at. The thermostat is new, so I don't think it's likely to be that. But today, in addition to the 'all or nothing' readings that I've been getting, it did start to wander up and down a bit (which is what it did when the thermostat was iffy).
More news tomorrow.....
...what a cliff-hanger, eh?