Both years I walked the Le Puy route, I started walking around 1 September. We averaged about 1 day of rain each week. It was hot (I live in Seattle, which has weather like London - your personal thermometer may vary) and I was in shorts and a breathable fabric collared shirt most days. One damp night in the highlands I really wished for a jacket. The second year, there was a real heat wave, and temps were in the 30s walking through the Basque country of France. September is very popular for walking, among the French, and you should check the open/close dates in MMD (many are only open from Easter to All Saints, and many gite owners take holidays at other times) as well as consider making advance reservations. Most French will call a day or two ahead for reservations as they go. The weather is lovely, there is all sorts of activity in the agricultural areas as the harvest is gathered in, it is a great time to go!