Well they'd spent several generations, at least, in France, had adopted the French language and christianity, and most likely the majority of French ways; copying the ways of the Frankish nobility no doubt. William the Bastard's mother was low born and may not have been of Norse background. But written information from that period (pre-Norman invasion Normandy) doesn't give up much information at all about his mum's side.
Norman Law is slightly different than English and Frankish/French law and has hints of a simpler background in things like 'common custom' being important guide, however it was codified way after the Invasion and thus not particularly indicative. This is a complex and layered topic a bit above my paygrade though.
The issue really is the question. Theres no definition of 'Norseness' that can be quantified, you cant say '87% Norse' or even 'fairly Norse' in a way that cant be argued with. Nor is there an easy way to list Norse vs Saxon vs Frankish 'styles' so that you can compare them. However, given the language and that William and his crew imported a fairly Frankish feudalish system in England then on the surface at least they seem quite Frankish. On a direct level the people who interacted with them, Frank and Saxons, would have been more aware of whether they 'felt' Frankish or Norse. But such information was rarely written down in such times and I'm unaware of any criticism or praise for their ways.
Sources, lots but Elizabeth Van Houts, The Normans in Europe.
I just realized I messed up the title, it should be
during Williams invasion
Without the added "the"
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