When Voldemort is about to break into Dumbledore's tomb to take the Elder Wand, the following happens:

"I shall join you in the castle shortly," he said in his high, cold
voice. "Leave me now."

Snape bowed and set off back up the path, his black cloak billowing
behind him. Harry walked slowly, waiting for Snape's figure to
disappear. It would not do for Snape, or indeed anyone else, to see
where he was going.

This seems to indicate that Voldemort did not want Snape to know that he was taking the Elder Wand from Dumbledore.

However, a month or so later, during the battle at Hogwarts we find the following dialogue between Voldemort and Snape:

"Why doesn't it work for me, Severus?"

In the silence Harry imagined he could hear the snake hissing slightly
as it coiled and uncoiled – or was it Voldemort's sibilant sigh
lingering on the air?

"My – my lord?" said Snape blankly. "I do not understand. You – you
have performed extraordinary magic with that wand."

"No," said Voldemort. "I have performed my usual magic. I am
extraordinary, but this wand... no. It has not revealed the wonders it
has promised. I feel no difference between this wand and the one I
procured from Ollivander all those years ago."

Here Voldemort jumps right into the discussion about why the Elder Wand doesn't work for him, without first explaining to Snape that he had procured the Elder Wand. This implies that Voldemort expected Snape to know about it, which would indicate that Voldemort had in fact shared this secret with him.

So what changed? Why/when did Voldemort go from hiding it from Snape to sharing it with him? It would be one thing if he only told Snape about it then when he was planning on killing him (so he wouldn't care if Snape knows), but he seems to have changed his mind and told him before this occasion.

The Dark Lord tells Snape of the Wand right before he kills him.

The Dark Lord only tells Snape of the Elder Wand when he’s already decided he has to kill him, and soon after telling him of the wand and his likely being its master, he does then kill him. The way he tells Snape of the Elder Wand is a bit disjointed and doesn’t give all the information in the most logical way, but it seems likely that’s the first time he’s ever told Snape anything about the Elder Wand, and only because he’s already decided that he has to kill him. He calls Snape away from the battle specifically to kill him to gain mastery of the Elder Wand.

“I have thought long and hard, Severus … do you know why I have called you back from the battle?”
_{^{- Harry Potter and the Deathly Hallows, Chapter 32 (The Elder Wand)}}

Then, he tells Snape how he sought and took the Elder Wand, to explain why he was going to kill Snape, before telling him that he needed to kill him. Though before that, he asked Snape why his wand isn’t working for him, that doesn’t necessarily imply that he expected Snape to know about the Elder Wand before then - at that point, he’s talking to Snape about his problem without having fully explained it yet. He’s telling Snape the story somewhat out of order - first that his wand isn’t working, then that he had a reason for calling him, third that he had the Elder Wand, and finally that Snape was the likely master of the Elder Wand and he realized he’d have to kill him to master it.

“Snape was not looking at Voldemort now. His dark eyes were still fixed upon the coiling serpent in its protective sphere.

‘I sought a third wand, Severus. The Elder Wand, the Wand of Destiny, the Deathstick. I took it from its previous master. I took it from the grave of Albus Dumbledore.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 32 (The Elder Wand)}}

It seems very likely that that’s the first time the Dark Lord tells Snape anything about the Elder Wand, or his search for it. He was possibly being purposefully vague at first, since he was originally trying to keep the Elder Wand a secret. It’s likely he only tells Snape as a way of explaining what he’s about to do, knowing that he has to kill Snape and the knowledge will die with Snape soon. Right after telling him about the Elder Wand, the Dark Lord tells Snape he must kill him to master it.

“The Elder Wand cannot serve me properly, Severus, because I am not its true master. The Elder Wand belongs to the wizard who killed its last owner. You killed Albus Dumbledore. While you live, Severus, the Elder Wand cannot be truly mine.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 32 (The Elder Wand)}}

He then tells Nagini to kill Snape, and Snape is then dead - he wasn’t left alive with knowledge of the Elder Wand for very long, since the Dark Lord told him only because he was going to kill him.

As soon as Voldemort decided to make use of it.

Snape could hardly have failed to recognize Dumbledore's wand, so he would certainly know something was up when Voldemort started using it. It would serve Voldemort's ends no purpose to attempt to keep Snape in ignorance as to why.

Why the secrecy to begin with, then? There are several reasons:

• If anyone had realized Voldemort was going to rob Dumbledore's grave, they might have tried to stop him, or to get to the wand first. Voldemort trusted Snape, but not that much.




• Even worse, what if the wand wasn't there? If it had already been removed, whoever had it would be put on their guard if they somehow found out that Voldemort knew about it.




• But most importantly of all, what if Dumbledore's wand was there, but wasn't the Elder Wand after all? Or what if Voldemort proved unable to use it effectively? That might make him look foolish, or weak, something which above all else Voldemort couldn't tolerate.

When it turned out that the wand was there, that it was as Ollivander had described it, and that Voldemort could use it just as well as his original wand, there was no further need for secrecy.