I assume techniques and technology used to cut umbilical cords have changed a lot over the years (and probably still vary around the world today). Is there some body of research into this topic?
Edited with Credible Source Contrary to popular belief, the belly button isn't formed by how the doctor ties off the umbilical cord, but is simply the left over scar tissue from when the cord naturally detaches from the body. Doctors are not able to completely cut off the umbilical cord attaching the baby to the placenta, which is why some parents still keep the shriveled up remained of the segment the was not cut off but naturally detached. So unless some cultures make sure that the entire cord is cut off and dig into the baby's stomach to detach it, belly buttons would have stayed the same throughout history. Source: The Fundamentals of Human Anatomy: Including Its Borderland Districts, from the Viewpoint of a Practitioner, Marsh Pitzman
The belly button isn't a consequence of how the cord is cut, but what remains after the "stump" left behind after cutting dries up and falls off. There would be no difference from modern belly buttons regardless of how the cord was cut - unless the cord was cut below the skin level of the infant, which as far as I know hasn't been a practice in any society. If that were the case, bleeding and infection would have a major impact on mortality.