Are there official / good unofficial rules for the new World of Darkness that would support actually and gradually playing through, over the course of a few stories, the transformation of a PC from mortal to a template? (I'd be most interested in rules for Mage.)
I do not think you need rules for this. The first thing to note is that for most of the supernatural, the change from mortal to being supernatural is generally a sudden discontinuity in their life. A sudden break with what came before. A vampire goes from being alive to being a vampire in the course of a few minutes. A werewolf is born a werewolf, and might find out in various ways, but even if they know from a young age that they are a werewolf, the first change will be a dramatic event. And a Mage's Awakening is normally portrayed as a something like a sudden flash of insight, a eureka moment that can be traumatic, euphoric, or simply awe-inspiring but generally fast.
So the transformation from normal to supernaturl per se is almost always fast. A few minutes or even less. But the rest, the acquiring of the skills, knowledge, and supernatural abilities that can all be roleplayed, and with pretty much no changes to the rules except perhaps in character creation and how advancement is done. And there you have a few choices.
What I would do is have the characters make normal mortal characters. All supernatural abilities, knowledges, merits, flaws, etc. are off limits but they can make some notes about what they would like to develop. Then, I would role play the transformation very early on and fast, but hand out the the supernatural parts over several game sessions. I would hand these out in lieu of experience, but far faster then they could earn them with experience, and I would hand them out based on the story but with player preference in mind. I would justify handing these things out so fast in two ways: First, there is a learning spurt whenever you enter a new field and they just entered a new field in a big way. Second, since these all flow from the story they are flowing from the story.
An alternative is to have them fully make the characters as though creating a new character of the appropriate type. But the supernatural parts are locked away. They then unlock the powers they already chose through roleplaying. This involves a bit more hand tying for the GM though since you need to find ways to justify what they did if needed.
Though I don't know of any official or unofficial Rules, I can see this be done for werewolf and changeling, but a Mage's Awakening is generally a radical and sudden change. A Mystery Play could probably be roleplayed over the course of a session, but I don't see it extending beyond that.
While not rules per se, there are guidelines in each splatbook detailing the minutiae of every template's -gradual or not- transformation, with roleplaying hints.
The Chronicler's Guide, on the other hand, offers alternative visions of the universe (such as one with 360 Supernal Realms, one for each combination of Ruling and Inferior Arcana) and the different forms of Awakening that happen within these variations. You might want to peruse this book for inspiration.
In the same book, you'll possibly want to take a close look at the alternative mode of play called "the Epistolary Chronicle". In this mode, the action is described through letters written by the protagonists. This could easily be used to relate the early apprenticeship of a newly awakened Mage, his bonding with a mentor, introduction to an order (even possibly his cabal) and other template-acquisition-related events before the game begins.