Northern Spain

You can get monthly average weather stats from each city's Wikipedia page, but looking at the day-by-day historical data will give you a better idea of the range of weather you may experience. For that, check wunderground.com.

Barcelona should be substantially warmer and drier than Bilbao and Oviedo, and I think there's a good chance of really nice weather in Barcelona in May. Oviedo is likely to be chillier than Bilbao, but you should expect some rainy days in both those cities, even in June.. Statistically, there would be about 40% to 50% more rainy days in May than June in all three cities.

Northern Spain is called Green Spain for the very reason it gets quite a bit of rainfall, especially when compared to the rest of Spain. June is better than May, is better than April when comparing both rainfall totals and the number of rainy days. Check out Oviedo's monthly statistics.
https://www.worldweatheronline.com/oviedo-weather-averages/asturias/es.aspx
Remember these figures are the monthly average, and if you are travelling at the beginning of June you may encounter 'May like' conditions rather than what would be considered 'typical June'.

In Logroño, La Rioja, during the 9th to the 11th of June we celebrate the day of La Rioja and St. Barnabas. Invariably there are a few cold days.

As they say here "Hasta el cuarenta de mayo no te quites el sayo", more or less "Until the 40th of May don't take off your cloak"

I have been to the area several times, twice in May and late April. This year, I was there until May 16. A couple of days we needed windbreakers. Other days, it was warm enough for shirt sleeves, and we waded in the sea. It was sunny every day except one in the two weeks I was there. May in northern Spain is beautiful, with the mountains covered with flowers. I think you would be fine going the third week of May.

Different parts of Spain and different weather. Barcelona will be different than Bilbao. There will be fewer tourists in May than in June, so I vote for May.