Arch your back, let your hair fall back but make it so none of those cold droplets fall on your skin anywhere other than your scalp. Owwww, a rivulet right down the back. Turn around, turn around. Maybe if you bent forward so that your hair falls in front of your face? Nuh uh. That way water just falls down your face. Icy cold and stinging. Oh Oh. Idea. Turn to the side and tilt your head just so. An angle so perfect that the water can soak your head but not trickle anywhere but straight down, not touching an inch of skin. Aaargh. Numb left shoulder now. Oh wait, was the water getting warmer? Skin isn't tingling anymore. Stretch out fingers, trembling hopefully, to the downpour. Yes, the water seems a lot warmer. Oh no, false alarm. Those are blue fingers. The water isn't warmer, the skin's wayyyy colder.
There is no non-fatal way of washing your hair when the water-heater's conked out.