You may already be using Via Michelin, a good tool for planning road trips. You have to keep in mind that it assumes no stops for any reason, and no serious traffic anywhere. You can enter intermediate destinations, like Sarlat, Bordeaux, and Tours, adding about six hours to the "unadjusted" straight-through travel time it shows for Nice-to-Bayeux.
With a week to drive the distance you obviously have enough time. Breaking down the destinations shows about 8 1/2 hours from Nice to Sarlat, 2 1/2 hours from Sarlat to Bordeaux, 3 1/2 from Bordeaux to Tours, and about 3 from Tours to Bayeux. That first stretch would be a very long drive, but you could break it with an overnight somewhere like Arles or Nimes or (slight detour) Albi, all worthwhile destinations. Two more nights each at Sarlat, Bordeaux, and Tours would be seven nights total. So it's doable.
But maybe not a very enjoyable trip. The Dordogne and Loire are worth more than the one full day you'd be giving them. Each time you change overnight locations you have to pack, check out, drive, check in, and unpack, so those travel days will give you less time for sightseeing than you might think. You could drive past a lot of attractive villages, caves, castles, wineries and such, but you wouldn't have time to stop at many of them. Nice and Bayeux are almost at opposite ends of Texas-size France. Do you really want to tackle them both, by land, in the same trip?