I'm a U.S. citizen living in Denmark. When I need to renew my passport, I mail it with the proper forms and payment to the US Embassy in Copenhagen, so I'm without it for the time it takes to issue my new passport. However, my new passport is issued in lightening fast time, only a few days or perhaps a week max, IIRC.
I would time it so I was very unlikely to need the passport during the short time I was without it. I suppose if an emergency came up where I suddenly needed to travel, the embassy would have a solution for issuing an emergency passport. But it really was done amazingly fast.
I did receive the old, canceled passport back as well (just to keep as a souvenir, I guess), but I don't remember if it came separately from the new one, or with the new one.
Susan