Pruning of Hazelnut tree

Depends what you want it to look like. Hazels can be pruned virtually to the ground. They will then shoot up from the base. This is known as coppicing, a traditional craft done by people who make hurdles.
It can be pruned to stay looking like a tree with one main trunk but you will have to remove the suckers. It looks wonderful if done like this. All old and gnarled, full of character.
However you intend to prune it do it in January/February.