I don't know the literature on cord cutting rituals cross-culturally, if
such exists.  Sorry.  What would be of particular interest would be to know
what the non-human primates, such as our closest relatives the chimpanzees,
do -- I've read from Jane Goodall's work (and actually seen also in films of
her work) that the mother chimpanzee carries the placenta (and cord) around
with the baby for several days until the cord dries up and separates on its
own.  I clearly remember one scene showing a mother carrying around a
stillborn baby, and the placenta gets caught on some bushes and the mother
goes back to unsnag it, gathers it up in her arms with the baby, and moves
off.  She didn't put the baby down for 3 or 4 days, when she finally gave up
on it.

Kathy Dettwyler