Well this afternoon I removed the dodgy looking caps someone had soldered
in. They crumbled into dust in the process; certainly not 500V rating no
way. Soldered in a replacement. Anyway, as predicted by two of our gurus,
no improvement.
So I've just pumped in 10VAC p-p sinewave @ 20kHz to the primary winding
and scoped all the secondaries. From these recorded voltages I shall now
work out the implied winding ratios at the various tapping points and see
if anything shows up as requiring further investigation (shorted turns
and whatnot). Never give up, eh?

'Laterz' as they say nowadays.