To 'tidy up' some of the answers above:
The Premium Service (where passports are issued on the day of an appointment at a Passport Office) isn't available for first passports (for either adults or children).
The Fast Track (one week) service is available for first passports (for both adults and children). The one-week time period normally starts from the date of interview (which might in itself, involve a wait of up to 3 weeks for an appointment) but, since there are no interviews for child passports, it would seem reasonable to assume that it starts from the date of receipt of the application. The fee is £87.
For normal passport applications by post (£46) or via the Post Office (£54.75) the only information offered by the Passport and Identity Service is that the process takes at least 3 weeks and that processing times may be longer between April and September. (I suggest that it would probably be wisest to allow at least 6 weeks).
The Passport Adviceline might be able to provide clearer guidance:
0300 222 0000 (0800 to 2000, Mon to Fri. 0900 to 1730 at weekends).
Chris