My well water is about 50F, give or take. First few minutes would be warm, as the basement is 70+ in the summer, but in the winter the water can be a bit chilly.
I usually shower in the evening and don't need the wake-up. If I shower in the day it's because I've been biking and the hot water feels good. Switching to cold (or at least cool) is something I should try, see if that cools me off faster.