I would not suggest hiking from south to north. unless you have personal or business reasons to be on the French Riviera at the beginning of your hiking vacation, or need to be in or near Switzerland at the end of it. If you think that by starting early from the south you can avoid snow in the passes, this is unlikely. The highest and most isolated pass on the entire GR5 system is the Pas de la Cavalle, at 2,671 meters (8763 feet) — (about 150 meters 500 feet) higher than any of the northern passes on the GR5, and it is only about a week’s walk from the Mediterranean.
The weather, in my opinion, should not be a reason to hike northward: The long climbs of the first few days will be in a hot-humid Mediterranean climate. Hiking from south to north provides an additional dificulty in that guide books and trail signs emphasize the opposite direction.