The advantage of cotton batting for baby quilts is that it gets softer and softer with each washing. I used Mountain Mist Blue Ribbon 100% cotton batting in a baby quilt many years ago, and after more than 15 years of hard use it feels soft as a cloud to the touch! The binding is worn, but the quilt is soft enough to rub against a baby's cheek.
Cotton takes longer to dry than polyester batting. For this reason, I would use only a thin cotton batting in a baby quilt. I wouldn't use Warm and Natural because it is a thicker cotton batting that is needlepunched through scrim. The thickness means it will take longer in the dryer, and the scrim means it will take many more washings before it starts to soften up. The initial stiff drape of W&N is more obvious in a small baby quilt than in a bed size quilt.
The advantage of a polyester batting in a child's quilt (I like Hobbs PolyDown) is that it will dry very quickly. For this reason, polyester batting is sometimes requested for charity quilts that will be sent to children in other countries (where they do not have dryers).