You seem to have a little anatomical confusion! The umbilical cord is attached to the placenta, which is "birthed" shortly after the baby. The baby wouldn't be attached to the mother for any extended period of time, cord cut or no cord cut. The umbilical cord can be cut or it will fall off naturally in a little time.

Unfortunately we don't allow "throughout history" questions, so I've removed this. If you're interested in the cord/placenta traditions of different cultures I suggest /r/AskAnthropology.