EDIT (you posted while I was beavering away on this!): Looks like the passport office and the FCO disagree. I’d do it your way if it was me. Good luck!
Your passport must be valid:
“You must hold a valid passport to enter France“
However, if you can get to London, Liverpool, Peterborough, Glasgow, Newport, Belfast or Durham you can get a same day appointment. So you may just have to delay your departure 12 or 24 hours. It takes 4 hours to turn the passport round. Phone 03002220000. You’ll need to take a completed application form with you, so download one off their site.
Details here: Directgov passport site