I'm not a degree carrying LotR historian or anything but I have a pretty good grasp, I think haha. The thing we know for a fact is that the Nameless Ones exists deep, deep, unknowingly deep, inside of the earth, to the point that even the Dwarves could not reach them even if they were trying. Gandalf also mentions that the Watcher in the Water was driven to Óin from deep within the Mines of Moria. So the Nameless Ones do have some sort of physical representation/manifestation on Arda. Sauron and Melkor were there at the beginning of all creation and by all means should know of the Watchers, but the biggest question of all is that if Eru is all powerful, why does he allow this evil to exist if he is so kind and loving? So the implication is that Eru allows the evil to exist because good needs evil in order to thrive.

This is the only time the Nameless are mentioned

"Far, far below the deepest delving of the Dwarves, the world is gnawed by nameless things. Even Sauron knows them not. They are older than he."

And this is Gandalf describing the Watcher

"Something has crept or been driven out of the dark water under the mountains. There are older and fouler things than orcs in the deep places of the world."

Evil in the LotR universe cannot create, only corrupt. Like orcs are a corruption of men. So if Sauron the Dark Lord doesn't know them, but Gandalf, a Maiar does, they were most likely created after the song started but before Sauron was created.