Dd1 I was private, I once waited 40 mins due to an emergency and had 1 appt postponed until later that day (had 1hour notice so was no issue).
public with my second the longest wait was 2 hours. I was seeing the obs rather then the midwives. One day I walked in to a lady screaming about waiting 3hours to see the midwives. I did feel a bit sorry for her when I was seen straight away.
I had to go to the clinic years ago after a miscarriage, was told there was at least a 4 hour wait. Didn't bother staying for that one!