All indications are that Neo has always been the One.
In The Matrix Reloaded, the Architect strongly implies that Neo has always been the One:
The function of the One is now to return to the Source, allowing a
temporary dissemination of the code you carry, reinserting the prime
Your 5 predecessors were, by design, based on a similar predication - a contingent affirmation that was meant to create a profound attachment to the rest of your species, facilitating the function of the One. While the others experienced this in a very general way, your experience is far more specific - vis-a-vis love.
The One is a designed function which carries Machine code. It makes sense that this design and code was given to Neo at birth, so Neo has always been the One.
Furthermore, when Neo is interrogated by the Agents in The Matrix, we see the room of TVs exactly like the TVs in the Architect's room.
Here is an image of the TVs from The Matrix:
And here is the view of the Architect's room from The Matrix Reloaded:
They are the same TVs, strongly suggesting that it was the Architect himself who was watching Neo's interrogation (and this was before Neo had even been unplugged or met the Oracle). However, the Architect would have no interest in Neo during his initial interrogation unless Neo was already the One.
As for the Oracle, she wants him to jump to the conclusion that he is not the One so that he would be willing to sacrifice himself for Morpheus. If she told Neo he was the One, Neo would believe he is more important than Morpheus and would not attempt to rescue him. Of course, it is the act of rescuing Morpheus that convinces Neo that he is the One, which was the Oracle's intention.
To understand this better, recall her demonstration with the broken vase:
Oracle: I'd ask you to sit down, but you're not going to anyway. And don't worry about the vase.
Neo: What vase? [looks around, brushes up against a vase, and breaks it]
Oracle: That vase.
Neo: I'm sorry.
Oracle: I said don't worry about it. I'll get one of my students to fix it.
Neo: How did you know?
Oracle: What's really going to bake your noodle later on is, would you still have broken it if I hadn't said anything.
This demonstration proves that what she says can influence Neo to do something he wouldn't otherwise do. She told him he was not the One in order to influence him to later rescue Morpheus, causing him to the conclusion on his own that he is the One. As she tells him, one must "know thyself".
...Being the One is just like being in love. No one can tell you you're in love, you just know it. Through and through...
Neo has to know he's the One, not simply be told by the Oracle that he is.
Finally, it's worth noting that the Oracle proves to us that she already knew Neo was the One before she "examined" him because she indicates that she knows Trinity likes Neo (although she doesn't mention Trinity by name). We find out later in The Matrix that the Oracle had told Trinity that Trinity would fall in love with the One, which of course she did. The Oracle knew all this beforehand.
In short, the Oracle lied to Neo about not being the One in order to influence him.