I just renewed my UK passport from the USA. You must apply online then send the old passport (and full color copy of any other passport!) to the UK.
Mailed application: Aug 12th (USPS Priority Mail International Flat Rate Envelope)
Arrived UK: Aug 19th 8:21 AM
New passport mailed: Aug 24th 10:45AM
New Passport arrived US: Aug 25th 12:25PM (DHL overnight from UK!)
Super fast, esp considering my UK passport had Syria, Iran, Algeria, etc visas...