This question is asked quite frequently on this forum. My response is always the same: it is by far easiest if the parents, baby, and nanny are all seated in the same cabin. It's not fair to other passengers to be constantly shuttling yourself and/or the baby between cabins.
I think you'll find four seats in business are less than two in first (moreover, you'll find that neither DL or AZ have an F cabin). If you meant business class (rather than first, as you say in your title), and cannot afford four seats in business, I'd take the previous poster's advice and consider four seats in premium economy.