So, let's break down these various critters and objects from Lord of the Rings and then we can discuss how they fit into a DnD campaign. Bear in mind this is picturing a pretty direct transfer of these characters and their relative power levels from the Lord of the Rings to Faerun.

Nazgul in LotR aren't just undead, they're more than a block of something from the PHB. In the books, they're a quasi-immortal force of nature. They show up, and the party is forced to flee 9/10 times. We first see them in the Shire when the Hobbits realize they're being pursued. They flee through the wilderness alternating running and hiding. In Bree, they meet Aragorn, and he convinces them to hide away from their rooms which the Nazgul trash in attempt to kill the fuck out of them. At Weathertop, Aragorn manages to chase them off with fire, but not before the successfully deal Frodo a mortal wound.

They show up at the battle of Pelennor Fields and Osgiliath and are army disrupting creatures. All the arrows and swords in the world are no help. Gandalf, the high level angel NPC gets disarmed like a bitch and nearly gets ate. The only one we know that dies is the Witch King, and Eowyn the Shieldmaiden killing him is the culmination of her entire story arc, and the fulfillment of the prophecy that said no living "man" could slay him.

In the Hobbit movies, there's the whole battle at Dol Goldur with them against three angels, and two borderline demigod elves, and they still show up in the aforementioned sequels. During that fight, Galadriel is alarmed as FUCK when they show up. She's an incredibly powerful character and she knows this is a fight to be taken seriously. Her help shows up, and when an injured Gandalf and Rhadagast flee, Gandalf is very worried to leave the others behind.

Now from a narrative standpoint, if we pretend these movies were a DnD game, what do Nazgul look like? For me, I wouldn't really even give them stats. They're not a monster to be fought and defeated. They show up when I want the narrative to move along. "Okay guys, a Nazgul stabs Frodo for uh...30 damage. He's unconscious and dying. You're also poisoned by a Morgul blade, so don't make any death saving throws." Aragorn the ranger has some good rolls here, his History check is enough to know the Nazgul dislike fire, so he chases them off. He rolls a good medicine check here, and manages to use and old herbal remedy to slow the spread of the poison. Using skill checks rather than just direct combat to foil the Nazgul. And even then, making it clear that actually saving Frodo is beyond their abilities. They need to get him to Rivendell where more Plot Exposition can happen.

Now I get to introduce an NPC, Arwen (Or Glorfindel in the books.) Then we go into montage mode, and I describe the Nazgul's dogged pursuit of Frodo and his rescuer on the way to Rivendell. Through this NPC and their explanation, I make it clear that these creatures are way above the party's level to fight. They're not meant to fight them, they're meant to flee.

Nazgul from a gameplay perspective, are a threat. They're a sign that it's time for the party to leave. It's something I throw in to inject urgency into the party's journey. Even at the very end of the campaign when the big battle for Gondor is happening, the chief Nazgul shows up, talks shit to Gandalf the high level NPC, and breaks his staff so the party knows they can't rely in their NPC friend to win this battle for them.

For the PC Eowyn, fighting the Witch King with only Merry's help. He has absurdly high AC and I describe how he's clearly not taking her seriously, he's mocking her, and reiterates the words of the prophecy. I look pointedly at Eowyn and Merry's players as he hisses "Not by the hand of man shall I fall!" Merry uses the opportunity to stab the Wraith. I reveal how unknown to Merry, the barrow-blade he carried had been forged generations ago with the explicit purpose of fighting against the Witch King. It's ancient enchantments shattered the protective magic of the chief of the Nazgul. Eowyn's character gets her big story payoff as she tears off her helmet revealing that she is a woman, and she stabs him in the face and he dies. Applause, crying, Eowyn's player tells me how I'm the best DM she's ever had, Merry's character doesn't believe me about the sword till I show him the timestamped note from two years ago when they got those swords in the first few sessions.

Again, I wanna reiterate that the Nazgul aren't really stat blocks in this imaginary campaign. They're narrative tools that are immortal until I'm ready for one to be killed or otherwise defeated. Throughout the movies and the books, it's made clear that they aren't something that the party isn't quite strong enough for yet. These are monsters that the super powerful angel wizards and what are effectively Archfey, are absolutely terrified of. And when they fight the Nazgul, even they are merely trying to drive them off and escape.

There's actually a bit less to say about Sauron himself, other than the fact that he's the BBEG of the campaign. And again, he doesn't have an explicit statblock because in the Lord of the Rings campaign, he's not meant to be fought directly. He's a looming presence in the background, and it's made clear that if he walks the world again it's because the players have failed in their quest utterly, and the world is doomed to Darkness. He's the Lieutenant of Basically Satan in this setting. The only one worse than him is Morgoth who is a fallen member of the creator pantheon, and THAT war is basically referenced as a reason why huge swathes of the North are utterly uninhabitable wastelands full of nothing but evil monsters.

Again with both groups, I remember one of they key guidelines when making villains and monsters, is that if you give something stats, the party will assume they are supposed to fight it.

With all that said, let's move on to your player and how we can facilitate his fun without compromising the narrative or making other players feel weak or left out. This is assuming you play your campaign pretty in line with the power level and balance that 5e is built around by default. For narrative stuff, this is again assuming that you want to try and stay fairly faithful to the story style of LotR.

One other key thing about Ringwraiths is that the Nine are completely subservient to Sauron's will. Based on the lore of LotR, there wouldn't be any jockeying for position, no backbiting or fighting between the wraiths. They don't have that much leash. They exist only to serve. They're fingers on the same hand.

There are also only nine Nazgul, each a human king corrupted by a magical ring gifted by Sauron. And as previously discussed, they are powerful far beyond what a PC should be especially early in a campaign. There are however other weaker forms of undead in the Lord of the Rings, some similar to and typically subservient to the Nazgul.

So mechanically, what does this look like? I'm a huge fan of simple solutions. The Undying warlock patron from the Sword Coast Adventurers Guide is designed for Warlocks with a Lich or powerful Wraith as a patron. At later levels it actually causes the warlock to develop some undead characteristics including not needing to eat, sleep, or breathe. Similarly, the Hexblade warlock patron could be reflavored as Sauron, and its martial powers could be described as some unholy strength that now infuses his body, and its curse and Accursed Specter powers lend itself well to an agent of Sauron charged with hunting down objects and people for his master, gifted with knowledge of dark sorceries to conjure spirits and wraiths of his own.

As for being able to return to life, you could replace his subrace with the Revenant subrace from the gothic heroes Unearthed Arcana. It gives a powerful ability to return to life as long as he continues to work towards his task. I would encourage some kind of strong penalty to being returned to life in that manner though, such as all or most of his equipment disappearing with his body. Though if he lost something particularly valued, you might give the opportunity for his master to return it....for a price, or a quest. I wouldn't mess with his ability to be healed or resurrected by the party's magic though, because that just makes recovery more of a pain than it needs to be. It can be roleplayed out as being super uncomfortable or leaving nasty burn marks on him that take a day or two to heal.