Presuming that you are leaving the country by air, you will need a passport to board your flight. You can obtain a passport almost immediately along with payment of the expedite fees and proof of your itinerary.

There is no "standard" time for replacing passports. You could send yours in today and have it back in 3-4 days or it could take 60 days. No way of predicting and, until you have it, you won't be traveling.