We are told that Sauron achieved control over the nine men because they wore the nine rings of power and the ruling ring controls the other rings and their wearers. So, if a Nazgûl would find the One Ring and find the will to use it, shouldn't they be able to break free of Sauron's influence?
Possibly, but it wouldn't have been to any avail.
Letter 246 outlines a hypothetical contest between Sauron and a Ring-wielding Frodo, and while a Nazgûl certainly has more innate power than Frodo, the more general terms of this outline would still apply:
Until Sauron himself came. In any case a confrontation of Frodo and Sauron would soon have taken place, if the Ring was intact. Its result was inevitable. Frodo would have been utterly overthrown: crushed to dust, or preserved in torment as a gibbering slave. Sauron would not have feared the Ring! It was his own and under his will. Even from afar he had an effect upon it, to make it work for its return to himself. In his actual presence none but very few of equal stature could have hoped to withhold it from him.
I've emphasised the last sentence because it's necessary to state that - no matter how powerful - a Nazgûl is still a lesser being than Sauron. They definitely don't fit into the "few of equal stature" category.
Recall that when Frodo claimed the Ring for his own, Sauron was immediately aware (Return of the King, Mount Doom):
And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made...
The same would apply to a Nazgûl, and the Ring must first be claimed before it can be used.
So if a Nazgûl were to claim the Ring, Sauron would come, and then the Nazgûl would be destroyed (and Sauron would get the Ring back).