About 6 years ago I was due to travel to Canada from Heathrow on a Saturday and woke up in the morning to find my passport had run out.
I drove like mad from Birmingham to the passport office in London and they did my passport in less than 4 hours.
I managed to catch my flight at around 3pm with a passport that was hot off the press.
I would have thought one of the major passport offices should be able do one for you if you can get there before 11am on Saturday.
Read this
http://www.ips.gov.uk/passport/apply-urgent-wi thin-two-days.asp