I have just had to renew my passport as it had less than six months left after I return to the UK. It's the same for everyone. My passport only took a week to arrive. According to my passport forms, by post is a standard 3-week service, then check and send at the post office which should take about 2 weeks,
Fast track service is a guarantedd 1-week service by appointment at passport office counters.
Then there's the premium service which is a guaranteed 1-day service by appointment at passport office counters. (For renewals or amendments only). Tel 0870 521 0410.