My children were born in Australia, where childbirth is considered a very "natural" occurence and everything (at least in my experience) is done in order for the future parents to live it as a natural, non traumatic experience.
I hated the whole thing in itself, especially because I desperately wanted an epidural but could not get one either time because it was too late, but I still felt extremely safe and protected. My husband was able to stay with me in a single room with bathroom for the whole time that I was in labour and delivering at the hospital and after the births as well. I felt we were living an intimate and rich experience that he could really share with me. He was very supportive and he cut the cords both times.
The only moment when he probably felt for a few minutes separated from me was when I first held my first born in my arms. He was saying to me:"Now there is three of us (Adesso siamo in tre - in Italian)", but I was in such awe that I could not answer him. I was feeling that very same emotion, that we had become a team of three, but I could not say a word for a while. I know he felt alone at that time, and scared in a way, but after a couple of minutes I was able to speak again and I told him how I felt. He is a very good dad, which makes me very proud to have chosen well...
One funny note: when our second child was born, my husband was a little more relaxed than the first time and he wanted to try the laughing gas himself. I did not let him though, because I was not letting go of that pipe for one split second, for any reason in the world.