I helped delivery the little girl in the picture when the doctor didn't believe my daughter was ready to push; so the nurse and I said, "Fine, stay at lunch, we'll do it." We did.
I was also present at the birth of her brother two years ago. I stayed for four days after but did housework and cooking and let them enjoy the baby for the most part. That is their bonding time and I wanted them to have it. They were sad to see me leave, which was a nice feeling.
I would offer my help and wait, like others have suggested.