You need to pay attention closely if you can be there.
They used to slap the babies on the behind to get them to take that first breath a cry. But as late as 33 years ago when the first my two were born the doctors said that was nasty for their first experience to be pain.
Both my kids started breathing at the moment they were confronted with the lights of the room in the form of a gasp, and neither made any crying noises. I'm not sure that it coincided with the cord cut, but it was my impression that my son beat the cord cutting to breathing first. The son was the second and I was less freaked by the whole event, plus he came quickly and I wasn't so much concerned about my wife who was at the end of 36 hours of labor. Daughter apparently needed a certain vbirth time and was holding for it. She almost got a C-Section.
Since I was sitting next to the receiving doctor I saw the birth of both. In the case of my son, the doctor claimed to be able to tell the sex by the shape of the buttocks. He said, "Oh you have another daughter!" I was at a slightly different angle, and I said, "Sorry, Doc, it doesn't quite look like that from here." Oddly they ran out of blue blankets and my son was put in a pink one at first. They were apparently having a run on boy births at the time.
That was about 30.5 years ago.