Baby it's Cold Outside.

Temperatures hit 24 Celsius in London this weekend, which is a rather balmy 75 Fahrenheit.  That's much cooler than almost anywhere in the United States right now, including the Pacific Northwest where overnight temps are still dipping into the 50s even though daytime temps are in the mid-80s.
June would not have been particularly warm in Shakespeare's time. He was born in 1564, right in the middle of the Little Ice Age and only a decade after major glacial expansion began.  There's a reason why they wore so many layers of clothes back then.