TL;DR Had Sauron gained the One Ring, Saruman would almost certainly have been tortured by him, although it is unlikely he would be used and would definitely not be made a wraith.
Tortured? Saruman himself certainly thinks so.
Gandalf had just escaped from Orthanc, and Sauron believed Saruman at least knew some info about the Ring. Saruman himself let on less than he actually knew to Sauron, so he was beginning to panic once he saw the Nazgûl at his doorstep. Being one of the most knowledgeable beings in Middle-earth would surely mean he had a good inkling of what Sauron would do to a traitor like himself.
Two days after Gandalf had departed Orthanc, the Lord of Morgul halted before the Gate of Isengard. Then Saruman, already filled with wrath and fear by the escape of Gandalf, perceived the peril of standing between enemies, a known traitor to both. His dread was great, for his hope of deceiving Sauron, or at least of receiving his favour in victory, was utterly lost. Now either he himself must gain the Ring or come to ruin and torment. But he was wary and cunning still, and he had ordered Isengard against just such an evil chance.
Unfinished Tales, The Hunt for the Ring
Also, consider what Gandalf says to Frodo about him being captured if Sauron won:
"You would have become a wraith under the dominion of the Dark Lord; and he would have tormented you for trying to keep his Ring, if any greater torment would have been possible than being robbed of it and seeing it on his hand."
The Fellowship of the Ring, Many Meetings
Frodo would be in the same situation as Saruman since both of them were opposing Sauron. One similiarity betwen their situations would be that Frodo had the Ring and Saruman had information of the Ring, and since Saruman didn't share that information with his supposed master it would make it just as bad as keeping the Ring himself.
Realistically speaking Saruman would still be considered an utter traitor in Sauron's eyes, not worthy of suffering a mere psychological torment such as seeing the Ring on Sauron's finger. Physical torment would be much more realistic: Sauron wasn't known for his leniency, even in the First Age.
Then Sauron smiled, saying: "That is a small price for so great a treachery. So shall it surely be. Say on!" Now Gorlim would have drawn back, but daunted by the eyes of Sauron he told at last all that he would know. Then Sauron laughed; and he mocked Gorlim, and revealed to him that he had seen only a phantom devised by wizardry to entrap him; for Eilinel was dead. "Nonetheless I will grant thy prayer," said Sauron; "and thou shalt go to Eilinel and be set free of my service." Then he put him cruelly to death.
The Silmarillion, The Lay of Leithian
Why put Gorlim to a cruel death when he could have left him there in utter misery of his wife's (Eilinel's) death? It was not in Sauron to give out mild punishments for treachery.
More Gandalf wisdom:
"No tidings of the battle will come to Mordor, thanks to the horsemen of Rohan; but the Dark Lord knows that two hobbits were taken in the Emyn Muil and borne away towards Isengard against the will of his own servants. He now has Isengard to fear as well as Minas Tirith. If Minas Tirith falls, it will go ill with Saruman."
The Two Towers, The White Rider
"He lives now in terror of the shadow of Mordor, and yet he still dreams of riding the storm. Unhappy fool! He will be devoured, if the power of the East stretches out its arms to Isengard. We cannot destroy Orthanc from without, but Sauron - who knows what he can do?"
The Two Towers, The Voice of Saruman
Enslaved or become a wraith? Not very likely.
Both Saruman and Sauron are Maiar. While Sauron is undoubtedly the most powerful being on Middle-earth with his Ring, it is highly not likely he would be able to corrupt another Maiar being - Saruman - in a way that Morgoth (Valar) corrupted some Maiar into the Balrogs.
It is equally more unlikely that Saruman would become a wraith of Sauron's. A Maiar is definitely not as inherently weak as a Man, therefore Saruman's outcome would be much unlike the Nazgûl's.
By 'incarnate' I mean they were embodied in physical bodies capable of pain, and weariness, and of afflicting the spirit with physical fear, and of being 'killed', though supported by the angelic spirit they might endure long, and only show slowly the wearing of care and labour.
The Letters of JRR Tolkien, Letter 156
It would stand to reason that Saruman wouldn't, or rather couldn't, be enslaved by another Maiar. Whatever trust Sauron had in Saruman was broken once he realized of the latter's treachery. I do not think Sauron would let him wander around free to do as he pleases, or edict a position of power (eg Mouth of Sauron gets Orthanc) to someone who is clearly untrustworthy.