Upvote:4
Unfortunately you cannot use the 1-day premium service (https://www.gov.uk/get-a-passport-urgently/cost - "Unavailable" in table for lost passports) Therefore, if you have lost the passport, you must use the Fast Track 1-week premium service available here: https://www.gov.uk/get-a-passport-urgently
You will have to receive the passport via the standard Royal Mail special delivery service and you'll have to be in to receive it.