It started nice and slowly like all important changes. It starts with a case, like always. John is listening to Sherlock list off his deductive reasoning, and just a split second before Sherlock says it - John deduces that the wife couldn't have done it. He only sees it because Sherlock has mentioned this sort of thing before. (The crime took place in a very muddy forest, the wife was found dead from suicide just a couple feet away. They had thought it was a murder-suicide but the wife had clearly been moved as she was wearing heels that didn't have any mud on them except from where she was laying. Therefore, she had been killed somewhere else and moved.)
Another time they're in a room looking at body, nothing new there. And John recognises the tie. It's one Sherlock had been experimenting on a couple of weeks. What demographics wore it; how long it took to strangle someone with (no humans were harmed); what fibres it left behind. At the time, John couldn't remember the answers to the questions. However, when Sherlock listed a few of them, well, they all came back and John made sure to remember them carefully.
It started to become more clear, to John at least, just a couple of weeks before the whole Moriarty/suicide thing. They walked into a crime scene, and as John looked around…. He missed things, but he could see what Sherlock was going to say, to deduce before it happened. The thing is…. John didn't notice. His Sherlock was there, why should he care that he could tell you what was going to happen next.
John Watson knew Sherlock so well he could mimic the genius. And in that way, John had become a genius of his own.