I doubt hazel milk or an equivalent was part of the hunter gathers diet at all.
I remember reading in "Ishi in two worlds" the story of Ishi, the last hunter gather in the US.
He disliked any cloudy soup. Tea, soups and stews all had to be clear.
He was really particular it about, he was almost suspicious of cloudy foods.
Rob
This might have been a regional thing - in Scotland at least Hazel coppicing was one of the first forms of farming -it was a heavily relied on crop and used in all sorts of dishes. I imagine a hot porridge of ground nuts would have been common, and so the possibility of using it as a nourishing drink is quite probable.
the problem of course with things like coconut/soya milk is, that while they are both very good substitutes for milk, neither of them are easy to obtain in the wild in the UK