I am traveling to London and Edinburgh (and Dingle, Ireland) in early June. What will the weather be like?

What should I wear? I am from the southeast US. It will be hot here in June.

We have everything...
You should bring nearly everything.

"If you don't like the weather, wait ten minutes."
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We've had a mild winter, and many plants are ahead of their normal timings but, while you should be free of the risk of snow by June, almost anything else is possible. And mostly on the same day.
Temperatures 70-80's F, or 50's.

You might need short-sleeve shirts and shorts, you're just as likely to need sweaters and a wind/waterproof.
(Ireland is Green because it's wet.)

London's less of a problem because there are lots of indoor attractions and sights, and the parks and a river cruise kept for the dry days.

There's acclimatisation, too: if you're coming from hot it will feel cold, even if it doesn't to the locals:
in Newcastle, in a blizzard of horizontal sleet, you can tell the Wuss...
He's the one wearing the long-sleeved T-shirt.

The BBC forecast a heatwave for the UK this summer, just last week. They didn't specify when, though. Should be at least reasonably warm at that time of year.

See link for Edinburgh info

The BBC weather forecasters also said that we were going to have a heatwave in 2007. Say no more!!
We like to think that our country's unpredictable climate is one of it's charms, but it is necessary to be prepared for almost anything. I'm sure there is a record of snow falling in England in June but, take heart, it is very unlikely.

London should see temputures in the mid 70s to high 80s usaly in June and July, Edinbrugh 60s and Ireland about 60s. Southern England and Wales is usually in the 70 to 80s this time of year.

Unpredictable and changable.

We had visitors from Australia come to Edinburgh a few summers ago and they only bought warm clothes with them so they had to all go out and buy shorts and tee shirts as it was hot every day they were here.
So bring clothes for a variety of weather conditions (especially rain), you won't need heavy coats though, it's never that cold in the summer.

It'll be between 60 - 70 in Dingle in June.