From asking doctors - I've gotten the same story that Mother Hen says: OH has to stay behind the curtain and can't cut the cord because the surgical site needs to stay sterile.
I was also told that as soon as the cord was cut, and the baby cleaned up and measured, that my DH could hold her by my side, but I wouldn't be able to until I was stitched up and out of surgery.
But given the other responses in this thread - I guess it differs by hospital, so probably best to ask your ob/gyn.