The Nazgûl fell for Sauron's trap
Evidently the men who became the Nazgûl wore their rings too much. They faded and became totally subservient to the Lord of the Rings aka Sauron.
‘A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the Dark Power that rules the Rings.
Merely posing the ring doesn't make you fade or become subservient the the Lord of the Rings, the Nazgûl evidently wore theirs and fell into Sauron's trap.
Nine he gave to Mortal Men, proud and great, and so ensnared them. Long ago they fell under the dominion of the One, and they became Ringwraiths, shadows under his great Shadow
Even when Sauron lost the ring he was still it's master, no one successfully claimed mastership of it.
Saruman manages to turn away the Ringwraiths and convince them he doesn't have the ring. This shows us that the Nazgûl are subservient to the Lord of the Rings - not Sauron personally.
‘It is not a land that you look for,’ it said. ‘I know what you seek, though you do not name it. I have it not, as surely its servants perceive without telling; for if I had it, then you would bow before me and call me Lord.
Gollum and the Ring
Gollum, while open to Sauron's influence was not subservient to it.
Gollum was captured in Mordor in the year 3017 and taken to Barad-dûr, and there questioned and tormented. When he had learned what he could from him, Sauron released him and sent him forth again. He did not trust Gollum, for he divined something indomitable in him, which could not be overcome, even by the Shadow of Fear, except by destroying him.
It seems keeping the ring but not always wearing it saved him from becoming a wraith.
No, not though he possessed the Ring so long, almost as far back as he can remember. For it was long since he had worn it much: in the black darkness it was seldom needed. Certainly he had never “faded”. He is thin and tough still. But the thing was eating up his mind, of course, and the torment had become almost unbearable.
One caveat though is the fading and domination failed when it came to Dwarves, probably due to their origins as children of Aüle. As no dwarf ever possessed The One it cannot be definitively confirmed what would have happened.