Take everything apart, give it a clean, grease and oil the bits that are meant to be greasy and oily, then put it all back together again. That often cures random creaks without ever actually diagnosing what the problem was.
A bed at my cousin's place in Charmouth was like that. I barely slept as I was trying to turn over without it creaking so I never relaxed enough to go into a really deep sleep and the creak woke me up if I did start to drift off
Thanks for all the advice (and abuse ). The plan is to swap pedals and wheels with those off Michelle's bike (sssh, don't tell her), and if it's still making a racket, replace the bottom bracket.
I've started to use it as a training aid - If I can get it to creak all the way around, I must have a really progressive pedalling action, putting in power all the way around.