I left in mid April twice. The first time, it was high winds and 5 C for the first couple days, then sunny, breezy, and 12-15 C for a few weeks. Then it started raining. We had some rain every day for the last two weeks, with about 18C for the high, 6 C in the mornings. Never saw a thing in O Cebreiro, it was foggy in mid afternoon. At the top of Cruz de Ferro, we didn't stay long, too cold and misty.
This past spring it rained out of the heavens for the first three days (epic flooding!), then the sun came out. We had ONE 20-minute shower (in Sahagun) after that. It was low 20's and glorious for the rest of the time. I walked the whole way in shorts and a T-shirt wearing sunscreen, although I started every morning with my fleece jacket until the sun did its thing. The rain started a few days later, and the June walkers got soaked.
You don't know what you are going to get, nobody does. Its not like southern California where the whether can be predicted years in advance, more like Texas where you might get a bit of everything, sometimes in the same day. You are also at 600-1500 meters in altitude most of the way. Be prepared for possible frosty mornings, and have a rain solution that you trust. With any luck, you won't need either, but you probably will. Have a fleece jacket for mornings and evenings, you'll need that for sure.
Buen Camino