Culinary school is a great place to start, but to become a private chef takes more than that. You need to work in the field for awhile, whether in restaurants or catering, to get experience and also contacts. I worked for a catering company to complete my apprenticeship to become a chef after school and then I hired myself out for various events, many of which were volunteer work just so I could prove I was able to run things on my own.
Do you cook meals where you nanny? And if so do you plan them or are you given a menu and/or recipes to follow? I ask this because if you are cooking at your job and you create the meals yourself you are doing the work of a private cook.
My advice is go to school, get work in the field and offer to cook for groups or individuals (they supply the food and you will do the planning and cooking) on a volunteer basis so you can get references for that type of work.
I now do catering (from private to large weddings), cake design and hire myself out to other caterers when the need help on a job. I can make my own hours, which is good with recent health problems.
Oh, one more thing. There are businesses springing up all over where they cook meals that people can pick up for dinner or for the freezer. You might want to look into a place like that to gain experience and even see if this is what you want to do.
Good luck!
. Made for a not so great time in wine country! 
