Based on the original seven books only: Voldemort would likely be the master of the Elder Wand.
It seems most likely that if Voldemort used Nagini to kill the master of the Elder Wand, the ownership of the Elder Wand would pass to Voldemort. To master the Elder Wand requires capturing it in some way from its previous master.
“Which is that the possessor of the wand must capture it from its previous owner, if he is to be truly master of it,’ said Xenophilius. ‘Surely you have heard of the way the wand came to Egbert the Egregious, after his slaughter of Emeric the Evil? Of how Godelot died in his own cellar after his son, Hereward, took the wand from him? Of the dreadful Loxias, who took the wand from Barnabas Deverill, whom he had killed? The bloody trail of the Elder Wand is splattered across the pages of wizarding history.”
- Harry Potter and the Deathly Hallows, Chapter 21 (The Tale of the Three Brothers)
Nagini was directly commanded by Voldemort to kill Snape - in this instance, Nagini was no different than any tool used to kill. Presumably the results would be no different than if Voldemort used a spell or a knife to kill the master of the Elder Wand.
“It cannot be any other way,’ said Voldemort. ‘I must master the wand, Severus. Master the wand, and I
master Potter at last.’
And Voldemort swiped the air with the Elder Wand. It did nothing to Snape, who for a split second seemed to think he had been reprieved: but then Voldemort’s intention became clear. The snake’s cage was rolling through the air, and before Snape could do anything more than yell, it had encased him, head and shoulders, and Voldemort spoke in Parseltongue.
There was a terrible scream. Harry saw Snape’s face losing the little colour it had left, it whitened as his black eyes widened, as the snake’s fangs pierced his neck, as he failed to push the enchanted cage off himself, as his knees gave way, and he fell to the floor.”
- Harry Potter and the Deathly Hallows, Chapter 32 (The Elder Wand)
Nagini was commanded to kill in Parseltongue, which it seems snakes must obey - basilisks can be controlled by Parselmouths.
“However, since Basilisks are uncontrollable except by Parselmouths, they are as dangerous to most Dark wizards as to anybody else, and there have been no recorded sightings of Basilisks in Britain for at least four hundred years.”
- Fantastic Beasts and Where to Find Them
Since Nagini was just a tool used to do the killing, she would not become the master of the Elder Wand.
With the Fantastic Beasts reveal included: Nagini would still not become the master of the Elder Wand.
At the time when Voldemort had turned Nagini into his Horcrux, she had almost certainly permanently become a snake by then. Over the course of their lives, Maledictuses will become the creatures they transform into permanently, and be unable to become human again.
“He whips open the curtains. There stands NAGINI in a snakeskin dress. Men in the crowd whistle and jeer.
Once trapped in the jungles of Indonesia, she is the carrier of a blood curse. Such Underbeings are destined, through the course of their lives, to turn permanently into beasts.”
- Fantastic Beasts: The Crimes of Grindelwald
Since she would no longer be a human, that she was human at one point is unlikely to be enough to make her the master of the Elder Wand, since at the time she was fully and permanently a snake. Since she would then be a snake, she would also presumably be compelled to obey any commands made in Parseltongue. It seems most likely that if Voldemort commanded Nagini, who would be no longer human and entirely a snake by then, in Parseltongue to kill the master of the Elder Wand, the ownership of the Elder Wand would still pass to Voldemort.