Is it possible to bring my Nanny with

Unless you are on a diplomat visa , you won't be allowed to bring your own nanny. There are many locals who can perform the job of nanny on the island.

Your employer would not be applying for a work permit for your nanny - they are employing you only. You could apply for a separate work permit for your nanny once you have your occupation permit in hand. However, as has been rightly pointed out, there are a lot of people in Mauritius available to perform this job and so the chances of your application being successful are quite minimal unless you can prove that there is a specific reason why your nanny is the only person to be able to care for your children and not someone else - and this justification needs to be to the satisfaction of the Ministry of Labour, who retains the full discretion to accept/reject any application for a work permit.