In reply to Scott K:
Online Visa application doesn't work if your passport doesn't have six months on it.
Phoning the Turkish Embassy just refers you to their website, which is ambiguous on the issue.
I phoned the UK passport agency at 8am and they do not do same day appointments, I could get one tomorrow morning, and should get a same day passport.
So choices now seem to be:
a) Get the flight and blag it at the airport.
b) Passport office tomorrow, pay for a new flight (£100) and catch that.
I'm so tempted by option a) but not sure if I'm brave enough!