Yes the same Hazel, Corylus avellana.
However, in the U.K. there are cultivated versions that are for nut production (as opposed to coppice for charcoal and timber), and also ornamental versions. Also there is Filbert (Corylus maxima) which is basically the same but the nuts are a bit different. Then there is the American Hazel Corylus Americana which you guys call Hazelnut.
I find it very hard to tell them apart and I think as gilsgils poured out the growing condition effect the timber more than the subspecies