Getting from England to France won’t be a problem if by car. I had an issue a while back when, the night before travelling, I realised my then-13 yr old duaghters passport had expired 3 months previously.
After a few panicked phone calls I decided to chance it – and sailed through on the way out, just waved a handful of passports at French customs.
Got a minor telling off on the way back though!
My advice – if you are dead set on going – phone the PP office, get things moving, go anyway, get someone to post the new PP on to you in France, and just stay there until it arrives!
Just take some folder red card to wave on the way out, they’ll never notice 😀