For what it's worth, which admittedly isn't much because I have no direct experience, I watched the temps of all the cities and towns along the CF closely every day in May/June 2022, and observed:
- regardless of the date, Roncesvalle seems to be consistently colder than any other city along the way. Temperature there was usually 5-10 degrees (c) cooler. So looking at the temps here can be a bit misleading with regards to the rest of the route.
- From mid-May to the end of June, I saw huge differences in temps, depending on the day and the location. But overall I found that most locations along the CF were at least 20 degrees (c), and often much higher in the afternoon.
- The night temps are significantly colder than the day temps at that time of year, or at least it seems that way to me. I noticed a big difference between nighttime lows and daytime highs in May/June 2022.
So if you're hoping for cooler weather, you might want to consider departing earlier than May 10, or ensure you walk early mornings. Once you get beyond Roncevalles you're likely to get relatively hot weather during the afternoon. That said, I'm sure there will be those that will report cold rain and even snow in May, so it's difficult to predict.
My situation is somewhat the opposite of you. I hate the cold and love the heat (within reason). I'm therefore planning on departing SJPDP on May 15, hoping to avoid most or all of the cold, but also avoiding the hottest of the hot days. So we're both considering leaving around the same time, and both hoping for the opposite weather
. Hopefully we somehow both get our wish.
Buen camino.