Why did the Nazgûl not try to reclaim their rings from a weakened Sauron together?

Question

It's mentioned in the Unfinished Tales that

Sauron’s “mightiest servants, the Ring-wraiths, who had no will but his own, being each utterly subservient to the ring that had enslaved him, which Sauron held.”

Even Gollum was bound to the ring he possessed once (the One Ring), was subservient to the ring's master, was enslaved to the One Ring. His will was also bound to the ring, but it did not stop him from betraying his master (actually the ring's current master, Frodo) to regain the ring.

So what would stop the Nazgûl from uniting and attacking Sauron to reclaim their rings? After all, Sauron did not yet have the one ring and did not have his full power. All the power that Sauron possessed at the time was through his servants (orcs, evil men, beasts and Nazgûl). Saruman had already been vanquished. As the witch king of Angmar was the commander of Sauron's forces, could he not lead a mutiny against Sauron? Is it that even in his weak state Sauron was undefeatable by all the Nazgûl combined?

Answer 1

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.

Dwarves

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.

Answer 2

The Nazgûl had no wills of their own.

... the Ringwraiths, who had no will but his own, being each utterly subservient to the ring that had enslaved him, which Sauron held.

The Hunt for the Ring (Unfinished Tales).

There are similar quotes in the Letters of J.R.R. Tolkien. I don't have this to hand, but you can find them by looking through other questions concerning the Nazgûl (e.g. here). Interestingly, not all of the men who became the Nazgûl were entirely evil from the outset:

And one by one, sooner or later, according to their native strength and to the good or evil of their wills in the beginning, they fell under the thraldom of the ring that they bore, and under the domination of the One, which was Sauron's.

Of the Rings of Power and the Third Age (The Silmarillion).

So, in the Second Age, Sauron controlled the nine because he had the master ring. According to the Tale of Years, Sauron is thought to have reappeared around TA1100, whereas the Nazgûl reappear around 200 years later. It appears that Sauron somehow restored them, but took away their rings to maintain control over them (since he no longer held the One).

Concerning Gollum, I think @Alarion is correct. Sauron influenced Celebrimbor when he forged the rings (except the Three), and then gave nine of them to men, specifically to ensnare them. He never intended that anyone else should hold the One, so there was no reason to give it the power to ensnare its wearer. Finally, it should be noted that hobbits are exceptionally resistant to the Rings of Power. In the Shadow of the Past, Gandalf says that

He proved tougher than even one of the Wise might have guessed – as a hobbit might.

This is why Frodo is the 'right' person to undertake the task of destroying the ring. Apart from the Three, rings of power corrupt their wearer by exploiting their desire for power. Since hobbits don't want power, they are very difficult to corrupt.