Hi Lucy and welcome to the Disney Parks Moms Panel.
It's lovely to be able to answer a question from someone in my home County of Lancashire!
Generally, I always allow approximately one hour when travelling from a Disney Resort hotel to any of the Disney Theme Parks, Disney Water Parks or Disney Springs (or vice versa) and usually aim to arrive 15 minutes prior to the time of any Advance Dining Reservations, just to be on the safe side.
In this particular instance because
Disney's Contemporary Resort is so close to the Magic Kingdom Park, I would recommend that you aim to be exiting the gates of the Magic Kingdom Park by no later than 12:40 PM on the day that you have a 1:15 PM Advance Dining Reservation at
Chef Mickey's.
You can either walk or take the monorail from the Magic Kingdom Park to Disney's Contemporary Resort. Walking will be the quickest option and will take you approximately 10-15 minutes. It might take slightly longer (20 minutes at most) on the monorail depending on how long you have to wait for the next monorail to arrive.
There is a fantastic transportation wizard which I like to use on the unofficial website www.our laughingplace.com.
It allows you to enter where you will be travelling to and from within the Walt Disney World Resort and then tells you various ways of making your journey, as well as providing estimated travel times. I would recommend that you add an extra 20 minutes onto the travel time provided to allow for time which may be spent waiting for the next mode of transport to arrive.
Wishing you a magical holiday filled with memories which will last a lifetime!
Emma