The IPS say they take 3 weeks to issue a new one (assuming you're renewing an adult passport). So just get down the Post Office and fill out the forms. You can pay extra, a fiver or so iirc, to get them to check it there and then to make sure the application is correct.
The IPS also offer a fast track 1 week service, costs more obviously. You'll need to get down to a passport office, there'll be one in a big city near you.
Finally, they also offer a 1 day service which costs even more. For both of these, you need to book and make an appointment. This is the number - 0300 222 0000.
But, 5 weeks will be plenty for a normal application.