Since the One Ring controls the Lesser Rings, if anyone had managed to master the One Ring they would gain control over the Ringwraiths as well

Characters like Galadriel, Gandalf, and Saruman who all have knowledge and experience with Rings of Power would stand a very good chance at usurping Sauron's place entirely. Gandalf and Saruman are even more obvious, since they're also Maiar, but it goes to show how all of Sauron's power relies on the One Ring, and why losing it is crippling.

Even his most elite minions are more beholden to the Ring than to him, since the Ring itself is what Binds them. It also explains why Saruman was willing to take the risks he did - with the One Ring it is very unlikely that Sauron could oppose him. Seeing as Saruman is a Ringmaker himself, he would gain control over it much faster than anyone else, leading to Sauron not even being able to use the Ringwraiths against him. And since they're both Maiar, a weakened Sauron is essentially an up-jumped Balrog.