Assuming one of Sauron's servants managed to recover the Ring, what exactly would Sauron do with it? He had no physical body to manipulate it and no finger to put it on. I guess he would have constructed a physical body for himself, but how? How would he get in "possession" of his Ring again? Just by a small enough distance between the Ring and the "Eye of Sauron"?
What would be the scenario of the event when one of Sauron's servants returns with the Ring?
The previous answers offer a good explanation of the book canon, but since your question is clearly inspired by the Great Eye of the movies, I'll make the distinction more explicit:
In the book, it's mostly a metaphor and a logo for Mordor, while Sauron has a normal physical body. He never appears directly in any scene of the book, he's only mentioned and described, which is why the movies were able to re-imagine him as a more concrete "monster", rather than the harder-to-portray psychic presence he takes in the books, without changing much.
So in the book, there's no question: he just would have put on the Ring, as anyone does.
The final cuts (and extended cuts) of the movies never actually comment on whether there's a physical Sauron in addition to the Great Eye (sort of like in the Wizard of Oz). But most interestingly, there was a deleted scene in the films that involved Aragorn fighting Sauron personally outside the Black Gate during the climax of Return of the King:
That scene ended up getting replaced by the fight with the Battle Troll, because the filmmakers decided that "those 10-foot monster fights" tend to come across as silly, and it drew attention away from Frodo's final showdown, which was actually the more important event in that moment.
But since Sauron had a normal body in the books and was supposed to appear with a normal body onscreen, I think we can safely assume that he had a normal body in the final cut of the film as well, regardless of whether we ever actually see it.
Therefore, both in the books and in the movies, Sauron would have simply worn the ring on his normal, albeit unusually large and claw-y and altogether evil, finger.
There has been some debate in the comments about whether the screenshot I posted is of Sauron, or the Mouth of Sauron. There IS a scene in the Extended Editions with the Mouth of Sauron, but I am talking about a specific thing they discuss in the Extended Edition documentaries where Aragorn fought Sauron himself.
Originally, Sauron would have appeared as Anatar, the beautiful disguise Sauron wore when he tricked the Elves into making the Rings in the first place. Aragorn would have let his guard down (this is the moment in the final cut when the Great Eye sort of beams-in on Aragorn and he slowly lowers his sword), before Sauron transformed into his evil form and attacked:
I was able to find this rough-cut footage of the scene, along with a few more stills:
According to Wikipedia, Gollum was personally tortured by Sauron. He tells Frodo and Sam about it:
Gollum tells Frodo that Sauron has, at least, a "Black Hand" with four fingers. The missing finger was cut off when Isildur took the Ring, and the finger was still missing when Sauron reappeared centuries later.
and according to The Letters of J. R. R. Tolkien:
...in a tale which allows the incarnation of great spirits in a physical and destructible form their power must be far greater when actually physically present. ... Sauron should be thought of as very terrible. The form that he took was that of a man of more than human stature, but not gigantic
So he did have a physical form, and could wear the ring if he ever recovered it.
I would add that while Sauron's original body was destroyed during Ar-Pharazôn's attempt to enter Valinor (when Númenor was destroyed), his spirit survived and fled back to Middle-earth. While it is not exactly described how Sauron was, I assume that he was very likely like the Nazgûl: When swept down by the flood or killed by Merry/Éowyn they left no physical body.
Sauron is likely like his servants more living in the Unseen world and needs magic to give his spirit presence; it is explicitly mentioned in the first book that Sauron gave the Nazgûl their garment to give their normal spirits a form to act in the material world. So as Daft said, he would have garb for himself enabling him to put the ring again on his finger.
Book: He does have a physical form, described by Gollum here:
`Yes, He has only four on the Black Hand, but they are enough,' said Gollum shuddering. 'And He hated Isildur's city.' —Lord of the Rings: The Two Towers, "The Black Gate is Closed"
So the answer is that he would put the Ring on his lost finger and thus restore it.
Movie: The Eye's pupil is Sauron's physical form, as shown in The Hobbit: The Desolation of Smaug:
So he would simply put the Ring on his finger normally.
You can easily imagine his physical form like those of the Nine Riders or Nazgûl in the film.
If you kill the physical form, the spirit remains and can reincarnate again later.
Well, Gollum says he saw a man with four fingers, a direct reference to Sauron, so we can assume he has physical or at least visible form. I like to think he just kinda chills in the tower and uses the eye to watch over Mordor.
If he got the ring back I assume he would put it on his other hand, or another finger, until the ring gave him enough power to regenerate his missing finger. He would try to use the One Ring to control the bearers of the three Elven rings, Gandalf, Galadriel and Elrond, controlling any who were too weak to resist or remove the rings fast enough and perhaps turning them into wraiths. Either way with the One Ring he would eventually collect the other three, cuz he wants them for their powers and stuff.
As for his next step, I really have no idea. Perhaps burn everything? Perhaps bend Man to his will? When he practically took over Númenor he created a Morgoth cult and killed people for giggles, till Eru sank the place, so if I had to guess I'd say he would do the same, and then either the Valar or Eru would step in and kill everyone. Or, alternatively, he enslaves all races and launches a full scale assault on the Valar, and rescues his old boss from the Void. Together they destroy the Valar and finally reshape the earth as they see fit, with darkness and volcanos and sick-nasty death metal album cover landscapes. Pretty rad. Unless your body isn't one that can handle lava and lack of air and the brutal awesomeness that is Morgoth's paradise.
His physical form would have been restored. It's said over and over he can't take physical form without the ring. The ring contains part of him that needs to be reunited with the rest of him in order to take physical form.
On a side note, I wish they had changed the final scene just a little. I would not have liked Sauron taking form without the ring, but he is a master of lies. Aragorn could have seen Anatar, and then Sauron, in his mind, as one final attempt by Sauron to make Aragorn think he has the ring.
