The official timeline doesn't claim BotW is at the end of all three, just that it's at the end of one of them but the decision on which has not been made. It could be they are leaving it up to the players to decide, or maybe eventually they will give it a placement.
Answering the question, a convergence of timelines cannot occur naturally unless the conditions are right for it, probably. Each version of Hyrule on the timelines at some point would have to be very similar, and have some of the same historical events. The outcomes of certain key events could be different, but the overall shared history would still be basically the same. This way a natural convergence can happen seamlessly and nobody would be aware of the change.
It's sort of like the Mandela Effect, where many people have a shared memory of something but it doesn't line up with the truth of the reality they are in. Either it's a widespread instance of false memory, or some would claim it is proof that there are multiple timelines and some people are shifting between them, the latter can never be proven though.
It's definitely possible that the downfall and child timelines could converge, events on the Adult Timeline are so majorly different though that it makes the odds of a natural convergence extremely low.
The next best thing is a divine convergence, either someone used the Triforce to merge the timelines, or the gods randomly decided to do it. How someone could be aware that there are other timelines is a big unknown that makes this an unlikely scenario. The gods randomly deciding to merge the timelines is sadly the best answer simply because it's the easiest one.