I live in UK and I have lost my Mauritian passport, how do I get a replacement?

Here's what the Passport Office says about lost passports:

A lost passport should be reported at the Police Station/Mauritian Embassy/Consulate nearest the place of loss and a police report should be obtained when submitting application for a new passport. The police report should be either in English/French language. In case the passport has been recovered, same has to be returned to the Passport and Immigration Office or to the nearest Mauritian Embassy/Consulate and must not be used for any travel.

So get the police report and fill out the application form for a new passport and send it by post.

I would advise you send the application form, police report and other relevant documents to a family member in Mauritius along with a written letter allowing them to collect the passport on your behalf. This will be faster than posting the application directly to the Passport Office or going through the Mauritian Embassy in England.

This article gives you a little bit more information.