I'll be using the pool for labour but I won't be giving birth in a tub
Babies can actually cry and try and take their first breath before they are completely born ( just head out) which means if you give birth under water your baby foul inhale some coming out .
This isn't true, there is as much chance of this happening in the womb as it is in a birth pool, babies only take a breath of air (or cry) when they are out of water, so this is when they are birthed on land from broken waters or in the case of water births when they have been pulled from the water after birth. Babies live in water for 9 months and don't drown themselves, they are getting their oxygen via the umbilical cord. The only thing you have to be really careful of is if you have birthed the head you must make sure babies head stays below water during birth, if you move around too much and lift your bum out and baby gets a breath of air then you dunk back down then yes this is really bad, but birthing pools are deliberately deep and MWs trained to tell you to keep your bum down. This was all explained to me when I asked if I could give birth in a bath, where due to the shallower water this is more of a concern.
So no a baby will not cry straight from birth during a water birth until you have pulled them from the water. Otherwise hospitals wouldn't endorse is as a method of birth! If it has happened I would like to see the statistics and I would bet apart from being incredibly rare it would also be a case of not being supervised properly and baby not being kept underwater until the end of the birth.