This post will be long and obviously contain some spoilers so, please be informed.
The recent conclusion of the ambitions made me think of Hesperidean Cider and its supposed connection with immortality. I posit that this connection is false, though we do know that there are, seemingly, some fruit with biblical/edenic connections. I will list my thoughts on the matter below.
Firstly, we can safely assume that all the mythical fruits we will discuss come from the Mountain’s Garden; and we know the Mountain’s effects result in extreme vitality, to the point of readily endangering more "mundane" life. Therefor we can surmise (dreadedly) that safe consumption of such fruits requires their brewing into an alcohol-adjacent drink, diluting their potency to a non-hazardous level.
Now, to the fruits themselves. The Fig, or at the very least, an Elder Continent fig analog, seems to be the fruit of the tree of knowledge; based off of the item one might get at the end of the Hearts Desire ambition and the half eaten fig in the Museum of Mistakes (suggesting that someone must have thought sapience itself is a mistake).
Immortality, or at the very least extensive longevity, is associated with peaches, with the Widow smuggling Peach brandy from the east. Up until now this connection was perhaps tenuous, with "eastern" peaches perhaps being merely highly restorative; but the Time ending of the previously mentioned ambition seems to point to to them being much more potent, with the Masters giving Peach brandy to the player character to ensure their prolonged life. This would mean "immortality" is only guaranteed so long as a regular intake of Peaches is maintained; which would fit, both with the Widow having to continuously smuggle more brandy and with the Empress having a notorious distaste of Peaches, indicating she might be forced to consume the self-same brandy herself and has developed an acute distaste of it.
But if the effects of Peaches and Figs is established, what, then, of apples? Hesperidean Cider is clearly made of them, the myth of the Hesperides clearly linking them and them alone as a fruit from which cider can be made. We know for a fact that drinking from the cider has healing effects; but but it seems text within the game itself refutes any real connection between merely imbibing the drink and enhancing lifespans. The text from the conversion of Broken Giant appears to debunk this directly; and furthermore, the text of turning in 3200 Scraps to the Capering Relicker has him tell us directly he drank the cider, and it gave him "lovely" visions which he followed, nothing more.
it is the Capering Relicker that might actually shed a light on what it is the Hesperidean Apples actually do. From other snippets of texts it seems that despite dealing in souls and other infernalia, Devils avoid the Capering Relicker assiduously; and that the Relicker smells oddly of freshly cut grass. This smell has in other places been mentioned as a telltale sign of the Garden, and while Devils might scorn or be disinterested in some souls due to their acquired flaws, active evidence seems odd, if not truly unique. Finally, we should consider the visions the cider gives the drinker: a melange of gnostic imagery seemingly emphasizing a naive and care-free primeval existence, and the accompaniment of a large cat. This last bit in particular is of some considerable importance, as it repeats both in Fallen London proper, in Sunless Sea’s Island of Cats, and most recently in Parabola, where a human and a cat walking together through the wilds is described by a Finerking as something that Is Not, i.e. something which has no representation in reality. And finally, the cat companion in these visions: we know from the King of Cats that there is a mythology amidst Neathy felines about a legendary Fall from Grace, resulting in the creation of cats as a diminished version of an original ancestor. We know cats are the designated defenders of reality from the Fingerkings - we can conclude that an early feline failed in their tasked and allowed primordial Man to be tempted, earning the ire of the Mountain and the reduction of its kin to a diminutive form.
Following this overview of what we know, I believe I have reached a satisfying conjuncture as to what it is Hesperidean Cider, and the fruit it derives from, actually does. I conclude that while the Fig grants knowledge, and the Peach grants life, the Apple grants Grace - meaning divine forgiveness. The Mountain may only be a hybrid Judgement but she is still, at the very least, a minor god in the FL cosmology; and the Devils being former servants of true Divinity might still be influenced by the disparity between this benign cosmic influence and their own tarnished nature that it forces them to avoid the Relicker as a vampire might avoid the cross. In addition this supposition interfaces well with the visions - for what more profoundly Is Not than Man under the full grace of God, a proud and unhumbled Cat at their side, walking the Garden without Original Sin?
I believe this means Hesperidean Cider allows the drinker to achieve partial forgiveness, by which they can enter the Neathy Kingdom of Heaven (the Garden) and sample its bounty without the usual restrictions, affording them potential access to both Figs and Peaches for a time without suffering ill effects from their consumption. This means that, if the Cider is to have any truly long lasting effects, a player must do as the Capering Relicker did and "follow the visions" - meaning make a sustained journey to the Garden, the origin of Man, where true forgiveness might be granted in full.