I plan to make a rpg setting and I want it to be as credible as possible.
That statement contradicts basically all of the rest of your premise.
Birds have very small brains, and think in a way which is utterly alien to us. Even if they started speaking to us we most likely wouldn't be able to follow their trains of thought (such as they are), or their instructions (if we play along with your premise of them being able to control us).
The idea that they might somehow "grow an organ that emits special brainwaves" is ridiculous. Additionally, humans would need to have "an organ" with which to receive and understand these "special brainwaves". We don't.
Ever heard the expression "If looks could kill ..."? If we could read minds, or otherwise be directly influenced by other people's brain waves that would be a big deal. We have no telepathic abilities whatsoever, so that is off the table.
Furthermore not all birds (even those of the same species) would suddenly gain this ability. Some isolated specimens would do so first, and maybe drive a human or two crazy. As the problem grew, people would start figuring out what's going on and exterminate the birds in question.
Birds cannot magically develop this ability overnight. Not even one bird could develop this ability overnight, let alone an entire group of them, or multiple species across the planet.
Once the threat was identified us humans are exceedingly good at killing things. Even if they had an amazing advantage over us, we would simply retreat and release a devastating biological weapon targeted at them. We would win this fight, hands down. (which brings up another point: there's no place on Earth devoid of birds. So I'm genuinely impressed that "Free humans survive by living in places where the birds can't go." since I can't think of such a place)
The last point I'd like to mention is that for isolated groups of humans to mutate to a noticeable degree you'd be dealing with a timeline of many thousands of years.
Answering Your Questions
For your story to be "credible" (as per your first sentence) a significant suspension of disbelief is required. I just couldn't go that far.
Since there's basically nothing short of magic which might bring this situation about it doesn't make sense to ask for a "realistic gap of time for the setting to occur". Simply make it up to suit your story, as you did the rest of your setting.
Similarly, the reason why "despite of the difference in structure between different human brains, everyone can be targeted" would be because the author says so.
There are simply no scientific reasons why that would be. You can't, on one hand, ignore science and biology, but on the other hand want scientific and biological explanations as to why your scenario is "credible".
Understand, I'm not saying that you shouldn't write your story! By all means, do so. However there's no good answers to your questions, since they're based on principles you're already rejecting in order to set up the premise of your story.