The umbilical cord will actually fall off fairly quickly (hours-days), as the circulation pattern within the child changes during and after birth. Usually, in the modern era, a portion of the umbilical cord is left connected to the neonate, maybe 2-3 cm, which then forms a stub which eventually falls off on its own.

Actually, as an aside, failure of the umbilical cord remnant to fall of is one of the things that pediatricians look for as it may be a sign of an [immune disorder](Leukocyte adhesion deficiency).

Hope this helps!

As a follow up, do we have any idea how long people have been cutting it off? I believe we have had sharp tools as long as we have been actual Homo Sapiens, but anyone is free to correct me.