Within the UK, HM Passport Office offer two options for urgent passport renewal: Premium and Fast Track.
Premium renewal is a same day service, but only for renewal and name changes of adult passports, where the existing passport is available and undamaged.
Fast Track renewal is a one week service, covering all renewals and changes, as well as new child passports (but not new adult passports).
Since your previous passport was lost, you cannot use Premium renewal. The fee for renewing a lost adult passport using Fast Track is currently £103.
To begin a Premium or Fast Track renewal, you'll need to book an appointment at a Passport Customer Service Centre. You'll need to take a filled-in paper application form (which you can get from a Post Office) to your appointment, as well as any required supporting documents including passport photos.