What are the dragon-like creatures that the Nazgul ride in the later part of the series called?
Anything in relation to the books or movie and out of universe answers.
Tolkien did not name the creature but based on different chapters of the books they were called by different names.
The Fell beasts, hell-hawks, and Nazgûl-birds, were names used to describe the flying creatures on which the Nazgûl rode after being unhorsed at the Ford of Bruinen.
When asked about the nature of the fell beast
Asked about the nature of the "steed of the Witch-king", Tolkien replied that the fell beast was not intended to be pterodactylic, but hesitantly acknowledges that it resembles a pterosaur and may have been a survivor of older geological eras.
The carcass of one of these animals (the one ridden by the Witch-king of Angmar) is twice referred to as a "fell beast" in The Return of the King.
Right before Theoden's death:
But thereupon Éomer rode up in haste, and with him came the knights of the household that still lived and had now mastered their horses. They looked in wonder at the carcase of the fell beast that lay there: and their steeds would not go near. But Éomer leaped from the saddle, and grief and dismay fell upon him as he came to the king's side and stood there in silence.
And again immediately after Theoden's death:
Men now raised the king, and laying cloaks upon spear-truncheons they made shift to bear him away towards the City; and others lifted Éowyn gently up and bore her after him. But the men of the king’s household they could not yet bring from the field; for seven of the king’s knights had fallen there, and Déorwine their chief was among them. So they laid them apart from their foes and the fell beast and set spears about them. And afterwards when all was over men returned and made a fire there and burned the carcase of the beast; but for Snowmane they dug a grave and set up a stone upon which was carved in the tongues of Gondor and the Mark:
There's no particular reason to believe that "fell beast" is being used as a proper name in this context, but most people seem to use it in this way because they aren't really named elsewhere.
Toward the end of The Fellowship of the Ring, the term "fell beasts" seems to be used to describe general nasty monsters that Frodo sees assaulting Elves and Men in Mirkwood in a vision.
But everywhere he looked he saw the signs of war. The Misty Mountains were crawling like anthills: orcs were issuing out of a thousand holes. Under the boughs of Mirkwood there was a deadly strife of Elf and Men and fell beasts. The land of the Beornings was aflame; a cloud was over Moria; smoke rose on the borders of Lorien.
The character Beregond seems to refer to the Nazgul's winged creatures as "hell-hawks" although he could be referring to the Nazgul themselves or even to the Nazgul/fell beast combo. This is from the scene where the Nazgul are chasing Faramir and Gandalf rides to his aid and chases them off with white light:
'Faramir!.... It is his call!' cried Beregond. '... But how can he win to the Gate, if these foul hell-hawks have other weapons than fear? But look! They hold on. They will make the Gate. No! the horses are running mad. Look! the men are thrown; they are running on foot. No, one is still up, but he rides back to the others. That will be the Captain: he can master both beasts and men. Ah! there one of the foul things is stooping on him. Help! help!'....
But there is no indication that "hell-hawks" is the proper name for the creatures either.
And behold! it was a winged creature: if bird, then greater than all other birds, and it was naked, and neither quill nor feather did it bear, and its vast pinions were as webs of hide between horned fingers; and it stank. A creature of an older world maybe it was, whose kind, lingering in forgotten mountains cold beneath the Moon, outstayed their day, and in hideous eyrie bred this last untimely brood, apt to evil. And the Dark Lord took it, and nursed it with fell meats, until it grew beyond the measure of all other things that fly; and he gave it to his servant to be his steed.
The description indicates it could be a prehistoric flying reptile. It is also possible that it is supposed to be the largest flying dragon left in the world in Third Age 3019. The description of its wings stretched between fingers indicates that its wings were more like bat wings than pterosaur wings. The description is unclear whether the witch King's mount was the same type of creature as those of the other Nazgul.
All in all, if the species should be given a name, I would suggest "terror dactyle".
I thought the beasts were called Nazgul, and the riders -- who were Men once that had fallen into Shadow -- were called Dark Riders (first on horses, then on the Nazgul). In LOTR: The Return of the King, the Dark Rider, named the Witch King of Agmar, when King Theoden is toppled and killed, says: "Do not come between the Nazgul and its prey!" I interpreted this as the Nazgul beast being the predator, and King Theoden its prey.
If course, John Ronald Reuel Tolkien knew best!