There does not seem to be any ancient text explicitly stating that these apples confer immortality upon anyone, although a case can be made that, thematically, they—as you put it—“at the least are somehow related to immortality.”
Precious Purpose
Contrary, indeed, to what the Wikipedia article that you mention says, neither is there any ancient literary instance of the apples being eaten or consumed in any way.1 The impression might be gathered here that this is not their purpose at all. Perhaps worth noting is that whenever golden apples occur in Greek mythology—and also, as it happens, in the Norse myth to which you've alluded in which the Jötnar (Giants) steal both the goddess Iðunna and her apples—there is no (explicit) mention of them being eaten nor of anyone attempting to eat them.
As silly as it might seem at a glance, the most obvious reason for this that I can see is that golden apples are—if understood to be described as solid gold (or anything close to this) rather than merely gold-coloured—are basically chunks of metal, which should be simply inedible, perhaps even to the gods.2 If this is the case then it would make much sense to conclude that they are a symbol more so than a practical foodstuff, even in the narratives themselves.
At the very least their function in the stories is that they are valuable on account of being (wholly or partially) composed of a precious substance. If for no other reason, Hera posts guardians over the fruit of her gardens because people have a tendency towards pilfering gold. I do think, though, that there is more to this than simply expensive fruit-shaped ingots. It seems to me that it is not the apples per se that have to do with immortality but rather the process of acquiring them which results in this eventual condition.
1. Juba of Numidia is said to have claimed that the Libyans called oranges “the Apples of Hesperia,” saying that it was Herakles who brought them into Libya shortly after stealing them from Hera's gardens.
2. Things get decidedly weird, however, when one considers that the Azteca apparently considered gold to be the excrement of the gods, which is literally what is meant by teocuitlatl, their word for gold.
These and Other Golden Apples
There are three instances in Greek mythology in which mortals interact with golden apples:
(1) A prince, either Meilanion of Arkadia [Arcadia] or Hippomenes of Megara, is enamoured of the renowned huntress Atalanta and is granted three such apples by the goddess Aphrodite in order to win the huntress's hand in marriage.
(2) Herakles is commanded by his slave-master Eurystheus, as the eleventh of his twelve athloi, “labours” or “tasks,” to steal for him the apples of the Hesperides from Hera's gardens in Libya.
(3) Prince Paris of Troy is charged with the responsibility of adjucating a beauty contest among the goddesses Hera, Athena and Aphrodite, the winner's prize being a golden apple.
None of these three men gets to keep the apple(s) that he has obtained, and all three eventually die. Herakles, however, becomes an immortal after his demise, which is a fairly common prerequisite for mortals' acquisition of immortality: they must die first, by some means or another. A few other differences are noticeable between Herakles and the other two heroes. Herakles takes the apples himself, by force, from the otherworldly location in which they have been planted by the gods, while Meilanion/Hippomenes and Paris are each freely given their apples while in the ordinary realm of mortals.
Herakles neither receives the apples directly from any divinity nor does he hand them over to any deity. He takes them directly from their tree and then presents them to his mortal slave-master, who gives them to Athena, who then returns them to Hera's gardens.3 Meilanion or Hippomenes receives his apples from Aphrodite. Another goddess, Eris, is the originator of Paris's apple, which ends up with Aphrodite after the beauty contest. According to Colluthus' Abduction of Helen, Eris' apple is actually one of those from the Hesperides.
3. Granted there is the version of the story in which the Titan Atlas (who here is the father of the Hesperides) is still an active character in the chronology rather than having been transformed, some generations earlier, into the Atlas Mountain Range upon whose slopes Hera's gardens are themselves located.
Herakles and His Assignments
The last three of Herakles' twelve athloi take place either on the border of the afterworld or in the home of the dead itself, which generally are all points of no return. These three missions are to:
10.0 steal the oxen of the giant warrior Geryones from the island of Erytheia, where the giant's friend and neighbour Menoites is the herdsman of Haides [Hades], the lord of the dead;
11.0 steal the apples of the Hesperides, the “Daughters of Evening,” who, like Geryones, dwell on the red edge of Day and Night at the world's western extremity (and also Geryones' island Erytheia, “Red,” is named after one of the Hesperides, who is the mother of the giant's herdsman Eurytion); and
12.0 capture Kerberos [Cerberus], the monster watchdog of the Underworld, whose brother Orthros, another monstrous canine, is Geryones' sheepdog.
To return from the afterworld by succeeding in these tasks signifies victory over death. This symbolism is not necessarily spelled out so obviously in the stories themselves but we do have the fact that Herakles is instructed by the oracle of Apollon [Apollo] at Delphi to become a slave of Eurystheus and perform all the tasks that he is commanded to. He is promised that after completing these labours, he shall receive immortality. Further regarding this, Dick Caldwell says:
Two of Herakles’ primary motives, the attainment of immortality
and the search for maternal nurturance, are embodied in the eleventh
labor. On one hand the magic fruit in the paradise garden, like the
golden apples of north European myth or the fruit of the Tree of Life
in Genesis 2, are a means to immortality; for the same reason the
streams of the garden of the Hesperides flow with ambrosia, the food
of the gods and the source of their immortality. On the other hand
both the apples and their location represent the idyllic existence of
the infant at its mother’s breast; the garden of the Hesperides, like
the garden of Eden and other paradise gardens, is a symbolic state of
abundant maternal nurturance. Herakles’ winning of the apples is one
of several attempts he makes to obtain this nurturance, the denial of
which was graphically portrayed in the episode of Hera rejecting him
from her breast.
In his 1902 commentary on Sophocles' Trachiniae, Sir Richard C. Jebb says severally that he perceives the original and now-lost meaning of the journey to the Hesperides and their golden treasure to be the attainment of immortality.
