The time you are considering is almost exactly the same time I went last year.
I have walked in both June and September. I think September (2016) was busier, (more pilgrims) or maybe just because it was a holy year.
We got exactly one and a half days rain each time while we were walking. Both times, thunder storms during some of the nights, which cleared up by morning. Both times the big cities of Burgos and Leon were freezing and everywhere else hot.
I enjoyed autumn, but loved spring. Birdsong, wild flowers, lush greenness everywhere - spring was magnificent. Wild poppies all through the Meseta.
But the best thing about spring for me, was the storks. They were in their nests with their babies. I was fascinated by them. Took so many photos of stork nests with occupants.
In autumn you can see the nests, but to see them with the birds, wonderful.
And in June, the cherries..!