The general advice for young babies is to use sterilised water for making formula, that is due to their immature immune system and because if they get sick they can get really sick really quickly. Whether the original source of the water is a drinking fountain, tap or spring populated by elves doesn't really matter but the sterilising is what's important.

As for the original question, it was about the UK (so TSA and water fountains are rather off topic) and I linked to the government rules on this.