I was with my wife in the delivery room, helping her 'push' the baby out for three of four of my children - it really was a struggle for my wife. You can see the veins on neck and temples - perspiring despite the aircond. I was glad to be present to welcome the child and yes I 'called' the 'azan' softly on the baby's ear almost immediately after the umbilical cord was cut. I always advised my Muslim friends to do the 'azan' themselves without waiting for some 'ustaz' or 'kiayi' to do it - and have honey (or dates) available for 'tahnik' (rubbing the roof of the baby's mouth with a bit of honey or crushed date - small amount only on the tip of your forefinger). My third child was born during the Friday Prayer so I was not present.
RE: accompany your wife when she gives birth to your child