Further to Dzug's answer:
When I last renewed my passport, I used the 'same day' service, at Peterborough. I was outside the office when they opened and I saw someone turned away because he didn't have an appointment. They gave him the phone number and told him to ring it, saying that they might be able to give him an appointment later in the day but they didn't think it would be possible. (I never saw him again, so I don't know whether he was seen or not. However, it would have obviously been better if he'd phoned before travelling tp Peterborough).
Chris