When Harry went to see Bathilda in the house, Bathilda (Nagini) saw Harry wearing the locket. Surely Nagini would have told Lord Voldemort. Unless Nagini did not know about Horcruxes, however later in the movie Voldemort says to Nagini, “the boy knows our secret”.
I can think of following reasons.
1. Voldemort completely underestimated Harry Potter
“One of us?” jeered Voldemort, and his whole body was taunt and his red eyes stared, a snake that was about to strike. “You think it will be you, do you, the boy who has survived by accident, and because Dumbledore was pulling the strings?”
Harry Potter and the Deathly Hallows, Chapter 36, The Flaw In The Plan
With Dumbledore out of the picture, Voldemort believed Harry went into hiding.
2. Voldemort was over-confident
“Accidents!” screamed Voldemort, but still he did not strike, and the watch- ing crowd was frozen as if Petrified, and of the hundreds in the Hall, nobody seemed to breathe but they two. “Accident and chance and the fact that you crouched and sniveled behind the skirts of greater men and women, and permitted me to kill them for you!”
Harry Potter and the Deathly Hallows, Chapter 36, The Flaw In The Plan
Voldemort believed that the Horcrux were well protected
3. He didn't know Harry and Dumbledore knew him inside out
Well, as you now know, for many years I have made it my business to discover as much as I can about Voldemort’s past life. I have traveled widely, visiting those places he once knew. Harry Potter and the Half-Blood Prince, Chapter 23, Horcruxes
“I think he would have envied anyone who had a key to a Gringotts vault. I think he’d have seen it as a real symbol of belonging to the Wizarding world. And don’t forget, he trusted Bellatrix and her husband. They were his most devoted servants before he fell, and they went looking for him after he vanished. He said it night he came back, I heard him.”
Harry Potter and the Deathly Hallows, Chapter 24, The Wandmaker
4. Horcruxes were well protected
Horcruxes were well protected either with trusted friends or kept at hard to find places. On top of that the Horcruxes were protected by charms.
Marvolo Gaunt's Ring:
I stumbled across the ring hidden in the ruin of the Gaunt’s house. It seems that once Voldemort had succeeded in sealing a piece of his soul inside it, he did not want to wear it anymore. He hid it, protected by many powerful enchantments, in the shack where his ancestors had once lived, never guessing that I might one day take the trouble to visit the ruin, or that I might be keeping an eye open for traces of magical concealment.
Harry Potter and the Half-Blood Prince, Chapter 23, Horcruxes
Salazar Slytherin's Locket:
“Not here, precisely,” said Dumbledore. “There is a village of sorts about halfway along the cliffs behind us. I believe the orphans were taken there for a little sea air and a view of the waves. No, I think it was only ever Tom Riddle and his youthful victims who visited this spot. No Muggle could reach this rock unless they were uncommonly good mountaineers, and boats cannot approach the cliffs, the waters around them are too dangerous. I imagine that Riddle climbed down; magic would have served better than ropes. And he brought two small children with him, probably for the pleasure of terrorizing them. I think the journey alone would have done it, don’t you?”
Harry Potter and the Half-Blood Prince, Chapter 26, The Cave
Adding to Vishvesh's answer, Voldemort believes that only he and only he knows the existence of the horcruxes and he is so immersed in his plans that he cannot "sense" his horcruxes being destroyed.
“Does Voldemort know when a Horcrux is destroyed, sir? Can he feel it?” Harry asked, ignoring the portraits.
“A very interesting question, Harry. I believe not. I believe that Voldemort is now so immersed in evil, and these crucial parts of himself have been detached for so long, he does not feel as we do. Perhaps, at the point of death, he might be aware of his loss . . . but he was not aware, for instance, that the diary had been destroyed until he forced the truth out of Lucius Malfoy. When Voldemort discovered that the diary had been mutilated and robbed of all its powers, I am told that his anger was terrible to behold.”
Harry Potter and the Half Blood Prince, Chapter 23: Horcruxes
Later when Harry and Dumbledore go to find the horcrux in the cave, Harry tries to stop Dumbledore from drinking the potion in order to retrieve the horcrux, Dumbledore says:
“I’m sorry, Harry; I should have said, he would not want to immediately kill the person who reached this island,” Dumbledore corrected himself. “He would want to keep them alive long enough to find out how they managed to penetrate so far through his defenses and, most importantly of all, why they were so intent upon emptying the basin. Do not forget that Lord Voldemort believes that he alone knows about his Horcruxes.”
Harry Potter and the Half Blood Prince, Chapter 26: The Cave
Not really. Nagini noticed the locket but she wouldn't have told Voldemort about it.
I'll answer from the movies since the films seem to be the basis for the question.
In the movie, Nagini (as Bathilda Bagshot) certainly looks intently at the horcrux. It's at 1:30 in the video.
The Horcrux seems to emitting an eerie orange glow for some reason. This isn't really explained in the film but it may be the Horcrux's way of trying to draw attention to itself in the presence of (yet) another soul portion. This is speculation, however.
That's not to say that Nagini noticed it as such. As Janus Bahs Jacquet rightly points out, we have no reason to think that Nagini recognised the Horcrux. She may have merely been momentarily distracted by an unexplained light. There's no mention of any great significance to this moment in the final screenplay, other than a mention that the horcrux "twitches" in Nagini's presence.
132 INT. BAGSHOT HOUSE - BEDROOM - SAME TIME - NIGHT 132
Harry watches as Bathilda moves closer, transfixed by her milky eyes. The Horcrux on his chest TWITCHES.
(Harry Potter and the Deathly Hallows - Part 1 Screenplay, Page 94).
It's maybe worth pointing out that, in the books, Nagini doesn't see the locket at all. It remains hidden under Harry's top, although it does jump when Nagini summons Voldemort.
In the film Voldemort doesn't come flying over to Godric's Hollow in pursuit of Harry and Hermione as he does in the books. So he and Nagini wouldn't have had an immediate opportunity to talk. Even if they had, there's no real reason why Nagini should make an effort to say, "Hey, guess what, the Potter boy had this freaky glowing necklace thing. Maybe you should check it out!".
Nagini saw the locket but had no reason to believe that it was of any consequence.
As for the piece of dialogue in the subsequent film, I think it may be reading too much into Voldemort's choice of words that he says the horcruxes are "our secret". Out-of-universe, this is just Steve Cloves's way of shoehorning Voldemort's thought process from the books into spoken dialogue. Since Voldemort's just killed a bunch of people he has no-one else to talk to, so Cloves has him talk to Nagini. It doesn't mean that Nagini knows all about the other horcruxes; she's just the character who was around at the time.
Voldemort: The boy has discovered our secret, Nagini. We must find out just how much he knows. We must return to our hiding places and see if the others are safe.
[Montage of images.]
Voldemort: And you, my friend, must stay close...
(Harry Potter and the Deathly Hallows - Part 2 Screenplay).
From an in-universe perspective, Voldemort is speaking in shorthand. He's internalising the urgent task which is before him: checking on his horcruxes. He's not expecting Nagini to keep up with him; he's simply thinking aloud. Nagini's main job, as with any supervillain sidekick, was to sit quietly and dote on her master - not to actually keep pace with his evil plans.
Nagini mightn’t have recognized the locket.
Though (in the movie version) it’s clear that Nagini does notice the locket, she didn’t necessarily recognize it as one of the Dark Lord’s Horcruxes. If she didn’t, she also wouldn’t then realize that Harry knew about the Horcruxes, and couldn’t tell the Dark Lord that he did. In the book, she doesn’t seem to notice it at all, even when she ends up pressing it against Harry while holding him for the Dark Lord.
“But nothing happened and he needed his hands to try to force the snake from him as it coiled itself around his torso, squeezing the air from him, pressing the Horcrux hard into his chest, a circle of ice that throbbed with life, inches from his own frantic heart, and his brain was flooding with cold, white light, all thought obliterated, his own breath drowned, distant footsteps, everything going …”
- Harry Potter and the Deathly Hallows, Chapter 17 (Bathilda’s Secret)
She noticed the locket in the movie, but didn’t necessarily know what it was on contact - Harry, who also had a piece of the Dark Lord’s soul in him, encountered both the diadem and the locket without feeling anything in their presence or recognizing them as anything unusual.
“And as he reached the end of the passage, the memory of a third stone effigy came back to him: that of an ugly old warlock, on to whose head Harry himself had placed a wig and a battered, old tiara. The shock shot through Harry with the heat of Firewhisky, and he nearly stumbled.
He knew, at last, where the Horcrux sat waiting for him …”
- Harry Potter and the Deathly Hallows, Chapter 31 (The Battle of Hogwarts)
It’s likely that Nagini could also encounter a Horcrux without recognizing what it was, meaning she’d have no reason to think or tell the Dark Lord that Harry knows about them. In addition, it’s not clear how much she knows about Horcruxes. She’d likely know that she’s one, because the Dark Lord would have done the spell to make her one, but it’s not clear if she knows about the other Horcruxes. The Dark Lord telling her “the boy knows our secret” could be because she knows about them (which doesn’t mean she’d have recognized the locket) or he could have been just talking to her without any real expectation she’d understand.