Does this Spell allow me to cast other spells with verbal components underwater?
There is not a yes or no answer to this, it is up the the DM of the game, sadly.
Do spells that require verbal components work underwater without the spell? Maybe, probably not I would rule it is like being gaged and therefore you can not make the exact pitch and tone of the words you normally use to cast the spell, but some DM's are cool with underwater casting without and water breathing.
Does the water breathing spell let you talk underwater, no it lets you breath, so again I would say it doesn't allow for verbal spell components.
Those are how I see this as the rules are written, but because of the vague nature of some of these rules and the wording of the spell, this might not be universally true.
Now, as a DM I dislike most spell requirements to begin with and would let my players get away with casting underwater if they had water breathing up. But I would understand that to me that is doing them a favor and no where in the rules does it say it should work.
Sometimes I miss the clarity of 4th edition, but there is a bunch of stuff I don't miss, so ehh.