C section panic

So I've heard stories where the mom isn't "all there" during the procedure and sometimes unconscious. I'm wondering how it works? Is the family allowed to go see the babies after the surgery, just like in a natural delivery? I know it sounds so selfish, but my fear is that I'll miss being able to hold them first. Excluding my husband, I don't want anyone else holding them before I do, even if I'm asleep for a few hours. Is that totally horrible?? I don't want to wake up 3 hours later and everyone's already seen them, held them, maybe even changed them! And I missed it all. I need to be the first. 💔