You can go to a passport office and get one on the day.
Edit: you are aware of that; I would not risk it, personally. I've used the premium service a couple of times in the past.
I renewed mine once by visiting the Passport Office when it was in Petty France, Victoria, I parked on the taxi rank in Broadway, by St. James Park station, and my present one the same way, but in the new building in Eccleston Square, Victoria, parking on the rank at the hotel in Bridge Place, (now the Double Tree Hilton).
It took a couple of hours, but I knew when I walked in that I’d walk out with my new passport.
Don’t try parking where I did unless you have a Black Cab.