The primary purpose of the human nose is to detect scents. This is an obvious statement, as is that the primary function of the mouth is to eat. The secondary purpose of both of these features is to breath. Now Merfolk who evolve underwater would have no need for the sense of smell (as it would be replaced with taste), so would the evolution of the Merperson favor the replacement of the nose for gills?
No. It wouldn't be favored. It would be required.
An organism with lungs and a nose cannot live underwater any more than you can. Before that point they have to evolve gills (including the "lungs"), at least in addition to land-based breathing.
edit: I think there was a misunderstanding. I am not claiming that any organism living in water needs gills (dolphins and other animals don't have them). Rather I am claiming that a human could not live in water without extensive modifications. While it would be possible to adapt the human organism in such a way as to allow it to live underwater without requiring gills, gills are the easiest solution as well as the one that constrains a writer the least (as in, if they have gills they don't need to have the merfolk surface all the time and can place them in deep water environments as well).