Last time we needed passports renewing we used the Post Office passport service.
Having first identified the Post Offices that did this function, we got the forms and filled them in. Next, we then took them to the PO where they checked they were completed correctly, they also did the photos there and then. The whole lot was given a reference number and sent to the Passport Office.
They quoted 3 weeks to us too, but we actually got them back in 2 weeks.
It was a really good service, the checking at the post office was thorough and helpful.