In the case that you find a passport that you reported missing, submit it to the U.S. Department of State at the following address: U.S. Department of State Passport Services Consular Lost/Stolen Passport Section, 1111 19th St., NW, Suite 500, Washington, D.C. 20036.
http://traveltips.usatoday.com/replace-lost-passport-fast-1697.html
An emergency, limited-validity passport will be issued to you if you need to travel quickly. In order to facilitate the replacement of your travel document, it is recommended that you maintain a copy of your passport in a safe but accessible location. Steps to Replace a Lost Passport. Step 1: Report Lost Passport.
http://www.us-passport-service-guide.com/lost-passport.html
Currently, there are two ways to report a lost passport to the U.S. Department of State. One, you can complete the Statement Regarding Stolen or Lost Passport, Form DS-64 (Opens in new window). You should fill out this form only if your lost passport is still valid.
http://www.us-passport-service-guide.com/lost-passport.html
If your passport expires in the next 9 months. You must renew your expired passport before you can travel. If your passport has been lost, damaged, stolen or your details have changed. If you’re replacing a lost or stolen passport, you may have to attend an interview and your application may take longer.
https://www.gov.uk/renew-adult-passport
If not, the U.S. Department of State will destroy the old passport. If you need to replace a passport while traveling abroad, contact the nearest U.S. embassy or consulate (see Resources). Diplomatic workers will evaluate your situation and instruct you accordingly.
http://traveltips.usatoday.com/replace-lost-passport-fast-1697.html
The same is true if your passport was stolen. Being the case, it will take the same amount of time as it does to obtain a new one unless you expedite your lost passport replacement. You can replace a lost passport while traveling abroad by visiting the nearest U.S. embassy or consulate general.
http://www.us-passport-service-guide.com/lost-passport.html
1 If you recover the passport after you have reported it lost or stolen, please submit it to the address listed above. When you submit it, if requested - we will cancel it and return it to you. If not requested, it will be destroyed.  Once a passport is reported lost or stolen, it cannot be re-validated.
http://www.us-passport-service-guide.com/lost-passport.html
How long it takes. It should take 3 weeks to get the passport - use a different service if you need the passport urgently. It can take longer if more information is needed or your application hasn’t been filled out correctly.
https://www.gov.uk/renew-adult-passport
1 They cannot be re-validated if the passport is returned to the owner. 2  U.S. Department of State: Lost or Stolen Passports. 3  U.S. Department of State: Form DS-64: Statement Regarding a Lost or Stolen Passport.  U.S. Department of State: Passports.
http://traveltips.usatoday.com/replace-lost-passport-fast-1697.html
Renew or replace your adult passport. You can renew or replace your passport at any time. It costs £72.50, or £82.25 through the Post Office’s Passport Check and Send service. When you renew your passport, the time left on your existing passport is added to your new one, up to a maximum of 9 months.
https://www.gov.uk/renew-adult-passport
how to replace a us passport
To replace a US passport contact the nearest U S embassy or consulate. To replace a United States passport, contact the nearest United States embassy or consulate.