If he used Nagini to kill its master, 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, since Voldemort controls Nagini. 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)
If Nagini was directly ordered by Voldemort to kill Dumbledore, she would then be no different than any tool used to kill. The results would likely be no different than if Voldemort had instead used a spell or a knife to kill the master of the Elder Wand. Voldemort believes Snape to be the master of the Elder Wand, and kills him using Nagini, so Voldemort at least thought that the Elder Wand’s allegiance would pass to him if he killed using Nagini.
“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)
If Voldemort commanded Nagini to kill Dumbledore
in Parseltongue, it would be even more clearly a direct kill by Voldemort. It seems Parseltongue compels snakes to obey those who speak it - 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
Additionally, Voldemort has even more control over Nagini than Parselmouths usually have over snakes.
“After an interval of some years, however, he used Nagini to kill an old Muggle man, and it might then have occurred to him to turn her into his last Horcrux. She underlines the Slytherin connection, which enhances Lord Voldemort’s mystique. I think he is perhaps as fond of her as he can be of anything; he certainly likes to keep her close and he seems to have an unusual amount of control over her, even for a Parselmouth.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
If Voldemort had Nagini kill Dumbledore, she would have been commanded by Voldemort to do the killing, so she would presumably be no different than any other method of killing, and the mastery of the Elder Wand would pass to Voldemort.