Many of the New England Nailors (and some who now spell their surname Naylor) are paternal descendants of the Cloutier Line of Chateau Richer, PQ. I am 10 generations removed (Paternally) from Zacharie Cloutier born in Mortgagne France about 1590. The surname was "anglicized" to Nailor when the family migrated to Vermont in the 1830s.

This is another bit of what Dick has to say.  I can't say definitively, the correct descension of the name.

It is NOT a variant of Naylor. To the contrary, there are a
number of instances where Naylor is a variant of Nailor. Particularly in
upstate Vermont. I would very much like to dispell the notion of Nailor as a variant of Naylor. It most certainly is not.

Now in my research, I've found several instances of my family of Naylor's being documented as Nailor, Nailer, Nayler, etc....
So I'm not really sure which is the "correct, first" anglicized version of Cloutier but I believe that with the illiteracy of the early Americans, the name was often just written as the writer thought it sounded.