How do you let you mum-in-law know that you will be calling the shots on the upbringing of your child (without sounding rude)?

Action speaks louder than words, unfortunately. Be firm in whatever you decide, and don't let her cross your line - especially in front of your child. Yes, it may come across as rude sometimes, but you don't want your child to have the impression that your words don't matter and he/she can go to your MIL for whatever you do not allow! You can tell her nicely and explain it to her first, and when situations arises, just set your foot down :) I do the same with my parents, I explain whenever I can and when they don't listen - I take action.