How long would it take to receive a passport with check and send service?Watch
A guideline for passport applications is definitely not a week - it's up to 6. It may be a lot quicker for a simple renewal, but if you need it in 2 weeks I'd go for the 1 week express option. You can find out about it on the official website.
As you've lost it, there may be some complications though. I have no experience with how long they take, but they have to be completely satisfied that it is actually lost and unrecoverable - not that you've just sold it to some for ID fraud purposes etc.