The answer is that your question is flawed, at least in part. Sauron did send a Nazgûl. As Grishnákh and the other Orcs are bickering, one of them says:
'I came across,' said the evil voice. 'A winged Nazgûl awaits us northward on the east-bank.'
-The Lord of the Rings, The Two Towers, Book III, Chapter 3: "The Uruk-Hai"
The "evil voice" is Grishnákh, of course. So Sauron did indeed send a Nazgûl along, possibly the one who had his mount shot out from under him, but probably not. Shortly after the passage quoted above, Grishnákh runs off from the main party of Orcs and then returns a bit later.
'Splendid!' laughed Uglúk. 'But unless you've got some guts for fighting, you've taken the wrong way. Lugbúrz was your road. The Whiteskins are coming. What's happened to your precious Nazgûl? Has he had another mount shot under him? Now, if you'd brought him along, that might have been useful - if these Nazgûl are all they make out.'
'Nazgûl, Nazgûl,' said Grishnákh, shivering and licking his lips, as if the word had a foul taste that he savoured painfully. 'You speak of what is deep beyond the reach of your muddy dreams, Uglúk,' he said. 'Nazgûl! Ah! All that they make out! One day you'll wish that you had not said that. Ape!' he snarled fiercely. 'You ought to know that they're the apple of the Great Eye. But the winged Nazgûl: not yet, not yet. He won't let them show themselves across the Great River yet, not too soon. They're for the War - and other purposes.'
-The Lord of the Rings, The Two Towers, Book III, Chapter 3: "The Uruk-Hai"
Sauron sent the winged Nazgûl, but ordered them not to cross the River Anduin, probably because the first winged Nazgûl sent out was immediately shot down by Legolas. Since Uglúk tauntingly asks Grishnákh if the Nazgûl has had "another mount" shot under him, we can probably assume that they already knew about the first mount being killed, and therefore, the Nazgûl who "awaits" the Orcs on the eastern bank of the river is a different one.
The Middle-earth Encyclopedia entry on Grishnakh says:
Grishnákh and Uglúk had an ugly dispute in which Uglúk cut off the heads of two Mordor Orcs who opposed him. Grishnákh reported these events to one of the Nazgûl on February 27, but the Nazgûl advised Grishnákh to pursue the Uruk-Hai and remain in the company with the Hobbits. On February 29, Grishnákh and his band of Orcs from Mordor once again joined the company of the Uruk-Hai. The Orcs and Uruk-Hai rested near Fangorn Forest.
And Thain's Book contains the following passage in the entry devoted to Nazgûl:
Sauron kept the Winged Nazgûl east of the Anduin at first. However, on January 9, 3019, an unexplained shadow passed over the Fellowship in Eregion (FotR, p. 299). Some have speculated that this was one of the Winged Nazgûl that crossed the river prematurely (HoME VII, p. 365).
According to a timeline published in The Lord of the Rings: A Reader's Companion, one of the Nazgûl met with Grishnákh on the eastern side of the Anduin near Sarn Gebir on January 26. Grishnákh was an Orc from Mordor who had information on the Fellowship's journey.
Grishnákh and a company of Orcs along with a Nazgûl attacked the Fellowship on February 23 as they were travelling downriver by boat. Legolas shot and killed the Nazgûl's steed with an arrow, but the Nazgûl survived. Grishnákh later joined forces with Uglúk of Isengard to attack the Fellowship at Amon Hen, and Merry Brandybuck and Pippin Took were captured.
Grishnákh met again with one of the Nazgûl on February 27. The Nazgûl told Grishnákh to pursue Uglúk, who was taking the Hobbits to Isengard. But Grishnákh was killed by the Rohirrim, and Merry and Pippin escaped on February 29. That same day, Frodo Baggins and Sam Gamgee heard the cry of a Winged Nazgûl overhead while in the Emyn Muil. It is possible that this was the Nazgûl who had been Grishnákh's contact.
As the previous answer suggests, Sauron deemed the Fell beasts too valuable to risk being lost before the war began in earnest. He sent one to oversee the capture of Merry and Pippin, but made them hang back a bit to stay out of danger.
We can probably assume that Sauron deemed the actual capture of the Hobbits to be too dangerous to get the Fell beasts involved, because Legolas had shot down the first Fell beast that was sent out, and the Orcs were carrying the Hobbits through hostile territory with the Three Hunters and the Rohirrim in hot pursuit; therefore, he sent Grishnákh to catch the Hobbits and bring them to the winged Nazgûl, who would be waiting in a safe place, and would then fly back to Mordor with the captives.
We might think of Grishnákh as a sort of Major or Captain, and the winged Nazgûl as a Lieutenant General. The General is in command, but too valuable to send to the front lines, so he stays away from the battlefield and gives orders to his lower ranking officers who are expendable and in the thick of the action.
Note: the quotes from Thain's Book and Middle-earth Encyclopedia are based on the same source, The Lord of the Rings: A Reader's Companion which the invaluable Tolkien Gateway describes as "one of the greatest recent secondary works on Tolkien's world."