Does Sauron have any use of the Witch King after regaining the One Ring? Yes.
Sauron wants to dominate and control the Free Peoples of Middle Earth. He did that by having the Rings of Power forged, gifting them to the lords of the Free Peoples, then forging the One Ring, through which he could control said lords and by extension all of the Free Peoples. It was not possible to do so all by his lonesome, hence why he had those Rings forged.
To rule Middle Earth, he had to address each of the three races:
- The Elves: By the end of the Third Age, the last of the Elves had begun to depart for Valinor. They depended upon their Rings to halt the fading of the Elves and sustain the might and splendour of their realms from Ages gone by. With the One Ring restored to its true master, the Elves had to give up on their Rings or become his slaves. They are no longer capable of defeating Sauron on their own by the end of the Third Age, so they are of little consequence to Sauron at this point, once he regains his full power.
- The Dwarves: Sauron had hoped to control them through their seven Rings but they proved resistant to his will. However, the Rings amplified their lust for mineral wealth, and that worked just fine for Sauron. The Dwarves removed themselves from worldly affairs, increasingly losing interest and concern for the fates of the Free Peoples. They are effectively taken out of the fight, and could be easily conquered once the rest of Middle Earth is lost to Sauron.
- The Men: These are the only ones capable of resisting Sauron from here on out. Had the War of the Ring ended with Sauron's victory and his One Ring restored to his possession, we would see the last Númenorean kingdom-in-exile of Gondor fall, and the rest of Man lacks the strength of mind and spirit to resist his Nazgûl, now at full power with Sauron fully rejuvenated. Just like back when Sauron first wielded the One Ring.
So, to achieve his goals, controlling Man is key. He does so by ruling them through fear - fear of his Ring-wraiths, the fallen kings of Man. And there is none more feared than the Witch King. There, hence, we can see the Witch King's value to Sauron in the ages to come.
Can Sauron reincarnate the Witch King? No.
The most important argument here draws from precedent: The story of Lúthien and Beren. Lúthien and Beren died and passed to the Halls of Mandos. By the beauty of Lúthien's song of grief, Mandos was moved to pity and desired to return them to life. Eventually, by the grace of Eru Ilúvatar, Beren was returned to life and Lúthien joined him in Middle-earth to live and die with him as a Man.
Another point to raise is that the Ainur all possess the power to create physical forms, albeit for their own spirits to inhabit in. Sauron could do that too. Thus far, there's no evidence that they could create new bodies for others - after all, the dead has no need for a new body. Sauron definitely has the means to create bodies, especially after he regains his Ring, the only question being whether he can do so for the Witch King.
Putting all that we know altogether, though, we can boil it all down to the following conclusions:
- The Witch King was not dead nor undead. He was a mortal Man who was kept unnaturally alive by his Ring. That meant that he suffered a mortal's death.
- The Witch King, as a Man, is not bound to Arda like Saruman or Sauron. So upon his death, he would (in his perspective) suffer the Doom of Man just like any other good or evil Man. The Witch King's spirit would have therefore departed Arda, never to return until the end of time.
- Even if he was still in the Halls of Mandos when Sauron regained his Ring, the latter has no power to bring the Witch King's spirit back - that power alone belongs to Mandos, a Vala, and even he does not possess the authority to do so - that belongs to Eru alone.
Therefore, the Witch King is not coming back.
There is a part in the Hobbit movies where Saruman remarked that there is no such thing as necromancy. I am not sure if those words were uttered by anyone in the books but by the above points, it stands to reason that this statement should be true nonetheless.
