For that area, the weather *should* be fairly decent - temperatures in the 60's, possibly the low 70's, but always bring an umbrella as you never know when it will rain/drizzle or shower. Of the places you mention, Paris probably will have the best weather. I've been to all the countries and cities you mention, including in June, and have had glorious weather. But that's not always the case. Just be prepared.
I would make the same weather comment as I make for Alaska or trans-ocean crossings. The weather is unpredictable.
answering as a Brit...expect everything.
surrounded by the sea.. our weather can change dramatically from one day to the other.
But June is usually a good time to visit. The temps may get to 70* But high 60's are the norm for that time of the year, and it is usually one of the least wet months.
And as you go North (Scotland) temps go down. I would expect temps around 60*
England, Ireland or Scotland.. I would always take a sweater just in case when traveling there.
plus the raincoat and brolly if you like them (I do not). I would also suggest taking a thin silk thermal set.
They will slip under your regular clothes and save you having to carry a lot of bulky clothes.
I would also take shorts