Gollum had the One Ring for a long time before the events of The Hobbit and Lord of the Rings. It deeply affected him, changing both his mind and his body. However, he did not become a wraith like the Nazgûl. Why? To understand the answer to this question, it is important to understand exactly how the One Ring works and who it affects the least. It is equally crucial to understand the purpose of the other Rings of Power in Sauron's plan. Every Lord of the Rings fan knows the story: the Elves have 3 Rings, the Dwarves have 7, and Men have 9. Sauron created the One Ring to use to control these other Rings of Power. The Elves, though, were too clever for him. They sensed when he created the One Ring and took off their Rings of Power.
Sauron then stole the Rings of Power from the Elves and gave 7 to Dwarves and 9 to Men. Celebrimbor made the 3 Rings of the Elves last, and Sauron played no part in their creation. There are, however, only 9 Nazgûl. This is because only the 9 Men who got Rings of Power ever became Nazgûl. There are no Elf-Nazgûl or Dwarf-Nazgûl. Of course, the Rings of the Elves were not directly made or affected by Sauron. But what about the 7 Rings of Power that the Dwarves got? Well, the Dwarves did not give in fully to the power of the Rings. At most, it made them a bit greedier for gold. Though Sauron tried his hardest, he did not fully corrupt the Dwarves.
The 9 men, who became the Nazgûl, however, are a different case. They were all fully corrupted and gave in entirely to the power of their Rings. This allowed Sauron to control them absolutely with the One Ring. And eventually, since they were only mortal, they became wraiths. Gollum, however, never became a wraith despite possessing the One Ring for centuries. To understand why it is essential to understand how the One Ring works and who it affects the least. The purpose of the One Ring is to control all the other Rings of Power. So, therefore, the One Ring is not really meant to turn anyone into a wraith. After all, why would Sauron do that to himself? No, the One Ring is meant to corrupt people and make them easier for Sauron to get on his side.
Gollum definitely suffered all of these effects of the One Ring. He became hard and cruel and completely devoted to the Ring or his "precious." Yet he did not become like the Nazgûl. Part of the reason for this lies in the purpose of the Ring, but part of it could lie in Gollum himself. He might not be a Hobbit exactly, but he is a similar creature. And Hobbits are not deeply affected by the One Ring. Though Gollum was certainly deeply affected. Yet it would not have been in the Ring's interests, or Sauron's either, to turn Gollum into a wraith. Firstly, he would not be much use in battle. And second, he wanted the Ring as much as Sauron did.
Therefore, the Ring and Sauron would've not seen the point in making Gollum like the Nazgûl when he could serve their interests as he was. Gollum also might have been like the Dwarves, not fully corrupted by the One Ring. Even though he might have seemed to be. There was still some of Gollum that was his own, and that might have been what saved him from a similar fate to the nine Nazgûl. Gollum also did not use the Ring a lot. It was too precious to him, and he did not see the need for it where he was in the darkness. That too might have saved Gollum from a worse fate. Though Gollum's story is definitely a tragic one, at least he was saved from being fully under Sauron's control like the Nazgûl. At least he didn't become a wraith, stuck in an in-between place all the time. Until the end, he retained a bit of agency, though he was mostly too devoted to the Ring for it to be noticeable.
Of course, not every Lord of the Rings fan will agree with this assessment of why Gollum didn't become like the Nazgûl after having the One Ring for so long. Some might believe the One Ring could have turned Gollum into a wraith, and that it would have been in Sauron's and the Ring's best interests. That's fine. However, it seems that Sauron wouldn't have wanted a soldier as pathetic as Gollum. And that Gollum too, was too Hobbit-like to be fully corrupted by Sauron. He might not have been a good creature, at all, but he did have a tiny bit of agency. And that is possibly what saved Gollum from becoming like the Nazgûl or any of Sauron's other servants. Gollum had a part to play in Lord of the Rings, but it was not in getting the Ring back to Sauron. It was, ironically, in destroying the Ring itself, thereby defeating Sauron.