The postal service, for renewing a child's passport, is meant to take a maximum of 3 weeks but that's not guaranteed. (The cost is �46).
If you use the Post Office 'Check & Send' service, the maximum processing time is meant to be 2 weeks, but again it's not guaranteed. (The cost is �46 + a �7 fee to the Post Office).
If you attend a Passport Office in person, you've got a choice of two services. You can either pay �81 for the 'Fast Track' one week service (which is guaranteed) or you can pay �94 to get your new passport on the same day (with just a 4 hour wait).
A child passport issued in October 2008 continues to be valid until October 2013, even when the holder becomes an adult. Additionally, any unexpired period of an existing passport (up to a maximum of 9 months) is added to the validity of a new one. So your passport would actually expire in March 2014.
There's a 24-hour official Passport Adviceline which can provide you with further information (and book an appointment for you at your nearest passport office):
0300 222 0000
Chris