When I was growing up in Coventry in the 1950s most factories worked on Good Friday but took the following Tuesday off as a holiday. There was good reason for that but unfortunately I now cannot find out what that was.
Not arguing here, just stating that because the January 2nd holiday here is relatively recent, and there would be few if any pieces of legislation (such as those which determine holidays) based on Scottish immigrant culture, our day-off is unlikely to be related to that in Scotland.
If you’d spoken to other Kiwis, Wendell Wagner, they could just as easily have told you NZ has heavy English influences, or Irish, or Polynesian or even Australian.