How can I get a replacement Mauritian Driving Licence if I live abroad?

This is the procedure you need to follow:

If you live abroad... You cannot get a driving licence duplicate unless you come in person. However you can be issued with a certificate (attestation) confirming you're the holder of a Mauritian driving licence. You will need to write a letter addressed to the Chief Commissioner of Police, stating your name, date of birth and address (both local and overseas) thereby giving permission for a family member or friend to collect the certificate on your behalf.

Source : Phone call to the police hotline (166).

For those residing in Mauritius... For a lost driving licence, you must report it to a Police Station. You will be issued with a memo (PF 77) for production at the Licensing Office when applying for a duplicate licence.

Useful link : police.gov.mu

Why would you need it now if you live in the UK?

You will need a memo and if you wouldn't come, a family will need to do it for you by a letter written by you I guess then, living in uk, you'll need an International license