Your premise is problematic
I am going to take two steps back and start by looking at what you want to achieve: a way to force humans to live under water.
Nuclear fallout does not really help you there, because of the way fallout works. Fallout is not some evil magical gas that floats around and makes surface life equally problematic everywhere.
In short: nuclear fallout is dust, not gas. The problem for you as an author with the fallout premise is that fallout does just that: fall out of the sky, as soon as there is some rain. That is how it gets its name, it simply falls out of the sky really quickly. This means that it is the surface that gets contaminated, not the atmosphere. And this contamination will not be uniform... it will instead be very uneven and spotty.
Nuclear fallout is harmful in two ways.
1) Direct radiation, we can call this "shine".
2) Ingested particles; i.e. dust particles that get into your body by breathing or eating.
Both 1 and 2 are solved in the most banal of manners: remove the dust, mechanically. Simply scoop it up and lift it away. Once that is done, prevent new dust from getting into your cleaned areas.
Doing this — cleaning up contaminated areas and make them fit for living and sustenance — is a lot easier and cheaper than moving down into the sea.
As evidence: this is how Hiroshima looks today...
So in short, you need another premise... and there is one:
The ozone layer is destroyed
The ozone layer is essential for us humans to live on the surface of the Earth. But even with the protection that the ozone layer gives us, ultraviolet radiation from Sol is still one of the leading factors to increase the risk of cancer(1), and very bad cancer at that: malignant melanoma.
The ozone layer absorbs 97 to 99 percent of UV radiation. So without the ozone layer, we could suffer 30 to 100 times more UV radiation on the surface of the Earth. This would pretty much preclude humans spending any time out in daylight.
Destroying the ozone layer is "easy" for you as an author. Remember that it nearly happened in real life. It was just that we caught the problem in time and fixed it. All you need to do is introduce a factor that does lead to the destruction of the ozone layer, and there you have a much more viable solution to forcing people to live under the surface of the sea(2).
(1) ...next after smoking, air pollution, viruses, and age
(2) ...or to simply become nocturnal.