The Lakes and Snowdonia were two places that left a big impression on me when the family visited the UK in 1988 (June): cold and wet and "out of season" like! Somewhere exists here Kodachrome slides of what I saw stepping out of the car in between "stream showers": an euphemism for chucking it down every 2 hours.
Snow is an insulator: you should not feel cold sleeping on it, but it will have mostly flattened or softened by morning. I've slept in the (Australian) alps at –6°C in a sleeping bag rated to –5, and it was effing cold!! Worse was being trapped in the tent by frozen zippers. Socks frozen, knickers frozen, beard frozen,. nose numb ... arrrrrrrgh—! The only thing that worked was the stove. Now it's Spring, warming up delightfully and I'm packing for a roadie.