Do Not - repeat NOT - go to the London passport Office. You will queue all day, with no certainty of getting through at the end.
Try one of the regional offices; a few years ago faced with the same problem I drove down to the office at Newport in Gwent, and received my new passport after about 40 minutes wait.
Get there before 2.00pm as they do (did) not accept applications after that time (worth checking), and make quite sure you have all the necessary documents with you - otherwise you will have wasted your journey.
Also worth asking before you go what the current waiting time is - I went in early January when things were quiet.
Otherwise its a long, long wait for postal service...
Go to the passport office in Glasgow (or if your still in englandshire, the nearest one)Tell them you need it NOW! and they'll do it for you there and then. (they did this for me a few years ago,get there early though)
The postal method is quite quick now - 1 to 2 weeks - but it is best to pay a small fee and get your local Post Office to check your form before posting. They also give you a tracking ref number so you can chase them if need be. I found the service was surprisingly efficient.
As I remember it, there is a guaranteed return time if you pay for it. And yes get the Post Office to check the whole thing for you. They charge a fee but it's worth it if you can't drive to the offices mentioned. There are other agencies like Lloyds Bank who will also do this.
It is a good idea to photocopy the "information" and visa pages and keep them somewhere safe - separate from your passport.
Then, if your passport gets lost/stolen/dropped or washed overboard, the photocopy is a great help to authorities in obtaining a replacement, getting home etc particularily if you are overseas when this happens.