No. We will mostly be dead. And if we manage to somehow not die, we won’t become mermaids all in one go. If water does take over our spaces really slowly, we are much more likely to become a kind of amphibious species first. This is assuming of course that the immersion is really slow. If it’s relatively fast, we will either perish completely, or be forced to find ways to float and live on the surface somehow. Think Waterworld.
As far as becoming an amphibian species is concerned, it might help to read up a bit about the Aquatic Ape Hypothesis. It is not a widely accepted idea, but it proposes that at some point of time in humanity’s evolutionary past, we spent at least part of our time under water. Again, this is not a theory, only a hypothesis. I am offering it here only as an aid to imagination.
Gills and fins are properties that aquatic species evolved over millions of years. Human beings in water might end up with variations of these organs given that long, or they may not. Mermaids are a fictional species created by people who did not know how animals acquire their shapes. We kind of do.