Recently I was thinking I was in a bind, and would need an expedited US passport. My passport isn't expired, but I was out of empty pages. I ended up booking a Passport Renewal appointment, and they were able to answer my questions fairly well.
They could request an emergency passport that would need to be approved by someone at the embassy. If it’s approved, it would be good for 1 year, and seems like it could take 1 week. It may of been up to 1 week, I can't remember exactly.
When I was there April 19th, they said passport renewals were taking around 3 weeks. I just had my appointment to renew my passport June 6th, and one person said 4 weeks, and someone else said 7-8 weeks.
Hope this helps!