Headcanons: On Servitude, and Selfhood
When Annatar chose the kings of men to whom he gave the Nine Rings, he chose carefully.* Each bearer had skill and abilities and personality traits he believed would be useful to him. Statecraft, warcraft, beastcraft, sorcery; lore of all kinds, and strengths both physical and mental.
Therefore, when at last the Rings' influence pulled their bearers into Sauron's control, he left their essences intact...for the most part. The Rings they bore made it impossible for them to rebel in any way; even rebellious thoughts were punished when they occurred. Sauron's own motives drove the actions of the wraiths, but it was by their own abilities that these motives were achieved. Thus, with his mind as the hammer and the Rings as his anvil, Sauron crafted nine perfect servants for himself.
However, it was also possible for Sauron to completely override the wraiths' senses of self and take over complete control of every aspect of their actions and their thinking. It was a kind of death, the Self supplanted utterly by the Other, but it was a kind of rest as well. Anything the Nine did during these times, they have no or very few memories of, and there are long gaps of nothingness scattered through their very long lives.
To what degree this overriding of self was achieved, and how often it was used, varied from wraith to wraith according to a multitude of factors including, among others, the wraith's own strength or unity of will and the specific errands or tasks for which the wraith was used most often. Some of the wraiths, like Ar-Dinenfaer, have almost no memories of their time under Sauron's control. Others, like Ji Indur, retain almost all of them.**
In the case of Uvatha himself, the degree to which his mind and will had been broken at the destruction of Kaemulothe meant that it was very seldom necessary for Sauron to override his will; Uvatha was as a dog, eager to please his owner at all times anyway. In addition, his work with the animals required a certain degree of autonomy to perform to any level of skill.
*Two of the rings did not end up with their intended bearers. Adunaphel took her father's ring after his death. Ji Indur stole his ring before it reached the intended bearer, whose name is lost to history.
**Ji Indur was a special case for two reasons. One, sailing a ship is a very difficult task and while Sauron could certainly have done so, he preferred to leave it to the expert. Two, the influence of Ossë meant that Ji Indur had a somewhat greater resistance to Sauron's powers.