Yes, we only have rules for teaching an apprentice a Virtue after the Gift has already been opened.
But it is also true that a character can take a Virtue as latent; the child has Gentle Gift, it just hasn't actually manifested yet. This is up to whoever designed the child's character sheet, presumably you, the GM, or some combination of the two.
We do not currently have rules for teaching Gentle Gift BEFORE the Gift has been opened.
In my campaign, I say that when a mecere finds a bloodrelated gifted child, they always see to that a Gifted Mecere gets it, because if someone else snatches the child, then that magus will be politely asked to give it away to House Mecere. If they do not, then said Magus will be forced to count on lost letters, being banned from help by redcaps etc. A little like a Maffia-organization. They will use threats and convince other Magi to coerce and harass the Magus.
Most Magi in the Order knows that you do not mess around with gifted children being of Mecere's blood.
The Gift itself can be latent; it does not always manifest at birth. Once it has manifested, however, one would expect that if it was Gentle or Blatant that this would also manifest. It would certainly be strange that a person's Gift would change character without the action of some outside agent (Twilight, Initiations, etc), and if I was the parent of Gifted children I certainly would not expect that to happen. Obviously apprentices develop virtues (including Hermetic ones) without being taught, as part of the abstract nature of their development, but there's certainly no good reason to just do nothing and hope they develop the virtue you want them to have (and some, like Mercurian Magic, are almost exclusively taught rather than gained naturally).
Probably the first response will be a Bonisagus arriving to claim the apprentice back from the snatcher, probably with Tremere and Guernicus hoplites right behind him. Then the child is quietly traded back to House Mercere.
I don't know, I could see the gift becoming blatant at puberty...
Careful, you're letting your prejudices show. 
Everything is for troupes to decide, sir. I'll point out further that the same paragraph that describes Magic Theory (vaguely) being the threshold for teaching a Hermetic virtue also says that "Teaching an Hermetic Virtue combines aspects of learning a Supernatural Ability (ArM5, page 166) with learning a Mystery Cult Virtue (The Mysteries Revised Edition, page 13)." That alone should suggest that knowing (Hermetic) Magic Theory isn't sufficient to being taught a Hermetic virtue, but in fact, that Opening the Arts is the "mystery" that must be unlocked to teach Hermetic Virtues.
The rules illustrate no need for the character to be "mature" enough to understand Magic Theory.
Nobody here claims, that Magic Theory is the only prerequisite for teaching an Hermetic Virtue by the means described on Apprentices p.40ff. This idea you found all by yourself, and I do not know how. An understanding of Magic Theory by teacher and student is just a prerequisite. Check for this also Teaching Hermetic virtues .
Without any common sense at all, Apprentices p.40ff does not make - yes - any sense. If you need rules here, look up ArM5 p.31 Early Childhood.
Just which Hermetic Virtues can be taught in a campaign by this method is a necessary decision each troupe must make. There was a lengthy discussion about this on this forum (see Apprentices: In my hands... - #66 by Matt_Ryan ), that we do not need to take up again here.
I give just two examples:
Cheers
You ask two questions here.
(1) What is the required relation of instructor and student of Hermetic Virtues on Apprentices p.40ff?
This is not certain. The structure of the chapters in Apprentices puts Teaching Hermetic Virtues as a subchapter into Hermetic Apprenticeship, implying that teaching Hermetic Virtues requires an Hermetic apprenticeship. But
puts this into perspective again. And on Apprentices: In my hands... - #66 by Matt_Ryan the author of Apprentices, Matt Ryan, appears to see no problem with shopping around for another teacher:
So this is utterly YSMV.
(2) What happens to existing Hermetic Virtues, when the Hermetic Arts are Opened?
Nothing. Only non-Hermetic Virtues form an obstacle here: see Apprentices p.35f Opening Arts with Supernatural Abilities, which is not cleanly worded, but clearly shows the intention.
Cheers
That's purely my interpretation of the situation and the rules...
Rules wise, there are a few points you might consider:
Initiation always requires active participation of the person getting initiated. Young Child won't have the knowledge and understanding to perform what is usually required in an Initiation script. The mother will have to create (or discover) a specific script for this occasion - considering Mercere House, it might be the only tradition who might have worked on that, but otherwise, the mother could get quite some fame to be able to initiate a child (or other people might be chocked by the proposal of submitting such young child to initiation).
Mechanically, as long as the purpose is not to give to future PC free major virtue, I would not bother with finding the exact mechanism, but more select a few adequate virtues or flaws to frame the story: infamous master or bad reputation (to reflect that the child was an experiment), some deficiency or affinity (maybe Vim since it is the form directly linked to the Gift), or chaotic magic (random side effect linked to the initiation). You can retroactively decide that this is the price to pay for such early initiation and in case somebody else would like to do the same, reconstruct a script requiring the acquisition of such flaw during initiation.
If the end result is a character with Gentle Gift on top of the regular 10 Flaws/Virtues and House package, then I would be more strict and follow the whole initiation procedure, very likely requiring acquisition of flaw. Otherwise, you will be setting an unbalacing precedent where it is possible to from now on for apprentice to start with a free major virtue. But even if the Gentle Gift is balanced with appropriate flaw, then you have a character with 13 points of virtues and flaws, which is also another can of worms you are opening.
... Unless, this becomes the focus of the campaign, with a feel of "Transforming the Order". Suddenly, the mother realise that with the appropriate script, baby/infant/toddler can have many virtues initiated very easily. Apparently without side effect (eg: acquistion of Major Flaw). Some Houses (Tremere, Bonisagus to start with) start a breeding program to pile up many Virtue, maybe even several Major Hermetic one in gifted children. Everything is fine, until much later during apprenticeship, they realise they were playing with fire and this new generation of promising apprentices is found to have to terrible flaw(s) and need to be restrained/quarantined/killed.
My intention was to give two NPCs (the twins) who wil probably soon go far away, and thus not contribute much to the plot have one of their 10 Virtues be Gifted. Not extra virtues over and above 10, and not allowing another Major Hermetic Virtue.
This is desired purely out of her Major Compassion to help those whom she can, rather than a power grab.
Bob
Consider also the possibility that it may not be compassionate to give the Gentle Gift in this circumstance. You're making their childhood "easier" which means that they won't learn the hard lessons necessary for adulthood, and it will make being a magus harder. We've all seen the fallout of permissive parents, and granting the child the Gentle Gift is probably a good example of that. Giving the Gentle Gift is also forcing your children to follow your character's path, not to create their own, more below. Of course, I play Tytalus magi fairly regularly, so the idea of overcoming challenges is probably burned into my mind with regards to magi.
They may not have a magical talent that lies in mundane realms; giving them the Gentle Gift then becomes a very selfish act, and not the best choice for the child. Consider that you might offer their future master the opportunity to Gentle the apprentices Gift at the right time, should he think it an appropriate virtue to pass on to his apprentice. Of course, the SQ requirement may well be higher, or you might have to pass on a Hermetic Flaw to make it happen, because it is likely that the master has either taught some Hermetic Virtues, or some inherited virtues have manifested. You might need to increase your Com, Teaching Ability and enhance your lab for the full +3 benefit to teaching.
I think we've established the technical limitations of your plan.
These are good arguments to present to your troupe, which will consider such issues.
As for the rules background for their decision: I believe, that we have now covered it here.
Cheers
A compassionate person would not accept that logic. They would act to minimize suffering, and Gifted children suffer from social isolation (or worse) while growing up, which leads to misanthropic magi.
Note that Jerbiton Magi would also consider that to be an odd chain of reasoning, as the Gift is something that makes magi... tacky.
Huh? Compassion is in the eye of the beholder, and compassion motivation may create wrong behavior. A compassionate person that is cognizant of the Gift doesn't value the Gentle Gift over other valuable Hermetic virtues. Childhood is a transitory period, the mature magus lives a long time.
No, Jerbiton magi consider studying the Arts for the sake of only more power and holing up in the lab for endless seasons to be tacky. Sufficiency dictates that they study the Arts, until they can do that what it is they wish to do and no more. Then they enjoy culture. Also, let's not say that Jerbiton magi have only magi with the Gentle Gift in them. Certainly some Jerbiton masters can't generate a sufficient teaching total to pass on their Gentle Gift (especially if there are already manifest Hermetic Virtues).
The Compassionate Flaw is pretty clear. "You cannot bear to see suffering in others, although you will happily drive yourself to exhaustion." That's not a character trait that says "Oh, some suffering NOW will be good for you in the future." If someone is suffering NOW, they will move to alleviate the suffering NOW.
If she is compassionate and witnesses her children suffering, perhaps she would just do this regardless of the consequences. It isn't really clear that the children are suffering, but it is implied that they might be, because her rationale is her Compassionate personality flaw. Let me further suggest, then, if she does indeed see her children suffering, she would just Open the Arts and Gentle their Gifts, regardless of the underlying consequences to her House/Order.
I'm not certain the children are actually suffering, though. And I have to say that, despite the flaw, I would really think that a character can envision actions now causing significant problems for their children later..
Compassion is always coloured by the person being compassionate.
Take, for example, the situation of a peasant levy being horribly maimed, losing both his legs and being in great pain as he lies bleeding on the battlefield; not dead, but certainly suffering.
Is it compassionate to alleviate his pain and save his life, or is it compassionate to swiftly end his life painlessly so he may pass on to his ultimate reward and not be a burden on his family?
The answer is 'it depends' - and it depends on more subtle nuance of the character than a single personality flaw.
As for the mechanical validity of initiating the Gift; if the story involved is fun or the parent's investment is significant enough, I'd have no problem with early initiation of the Gentle Gift.
It doesn't really matter if the child is intended to be a PC later on or not; characters developed through actual saga play from birth shouldn't be bound by balanced virtues/flaws at the point they complete their apprenticeship. Their virtues/flaws should instead be a reflection of what has happened to them through play.
Yes, it means that it is possible for a player to swap to an 'optimised' PC after ~15-20 years of saga play, but they're still going to be 15-20 years younger than the other PCs.
Too true. In fact, both responses are compassionate, as they both deal with the suffering, though one is a very practical/ruthless way of doing it.