Update: Read my follow-up post, Timeline Merge Theory.
Zelda fans right now certainly will go crazy over timeline theories again. It's obvious that Breath of the Wild does take place long after the Zelda series as we know it. We have entered an age of technology and we see the remains of past places, like the Temple of Time. So, the real question is, which of three timelines will make a wild breath into the future!
My personal favorite before the E3 presentations was the Downfall Timeline, because of several reasons. The world of Zelda II - The Adventure of Link has been the largest in Zelda yet and it made sense to expand from there. They also wanted to go back to the roots, so placing the game after the NES Classics would be meaningful. It also fits the idea of a ruined Hyrule very well with a rusty Master Sword back on the pedestal that we knew from A Link to the Past, as well as the idea of resurrecting the Hero, where the Downfall timeline is where Link's death actually has outcomes. Ganon also gets mentioned as the antagonist (it seems like he was imprisoned inside Hyrule Castle) and he has been prominent in the Downfall timeline mostly.
But Ganon appeared in Four Swords Adventures as well and we get to warp in Wolf Link as a companion (this part is probably non-canon, though), so the Child Timeline might be also a candidate. We even see structures that are reminiscent of the Hyrule from Twilight Princess, like the Great Bridge of Hylia. What speaks against this specific timeline is the fact that we still have the Temple of Time in the new game in some shape, while it was completely ruined in Twilight Princess.
The Adult Timeline gets supported by the Koroks, who seem to be very close in appearance to the Wind Waker. Those might even be the same guys, who have planted the trees all over the Great Sea in the hope that more land will grow. And these could be the lands, where the Great Sea was ages ago... But we already got a Hyrule there and Breath of the Wild shows a remarkable lack of rails so far. And again, the Temple of Time shouldn't be here anymore, if those are the new lands after the era of the Great Sea, unless the Temple survived somehow and they simply decided to move the Master Sword elsewhere. But the Temple of Time pretty much points towards the Downfall Timeline and it's possible that the Koroks simply existed there as well.
If it's in the other timelines, due to the Temple of Time the game would have to take place either before Twilight Princess or The Wind Waker, which wouldn't be all that satisfying. Nintendo would have to bother with connecting the games correctly or they would leave a mess behind. And the Sheikah technology would get wiped away afterwards, while it looks like a promising new feature for the future of Zelda. I think, this game would be best at the end of the timeline(s), where it can freely lead into a new era of the Zelda series, instead of being just another note within the timeline. And that's probably best after the NES games. All other games since the classics went either backwards or sideways. And now it's time to move forward.
But I have a bold prediction: why not all three timelines? Already in Skyward Sword the technology to mess with time was displayed by the timeshift stones and the Gate of Time. The latter was even related to the Sheikah and the Goddess Hylia. I suppose the Sheikah or Hylia, who could even turn out to be the (new) Goddess of Time, could take a similar role to Cia in Hyrule Warriors, overseeing all the times. Now, maybe they saw a reason to merge the timelines back together. (I know that theoretically this would be impossible and pure nonsense, but so is time travel into the past.) Maybe a great evil (Ganon or even Demise in some other form) arose in all three timelines and to contain it they fused them all together, which created the new Hyrule in Breath of the Wild, where elements from all the Zelda games are coming together.
The things that existed in all three timelines got fused and now show traits of all their versions: Hyrule, Ganon and the Hero. This would explain, how this "Calamity Ganon" could take so many different forms, because he's essentially Demise's reincarnation from all three timelines. And this would explain, why get to meet Koroks on top of the Temple of Time.
I could already picture, how in the story the timeline gets displayed on one of the Sheikah screens explaining the convergence, displaying Ocarina of Time and Breath of the Wild as the polar opposites in the timeline...
The timeline got split into three, much like the Triforce. And to further this symbolism it would be meaningful to reunite all three universes at some point, much like the Triforce.
That might be a crazy idea, but it would also be an interesting way of "rebooting" the franchise without completely ignoring its previous installments. Instead it would fuse all the different universes back together into one experience. If Breath of the Wild really is supposed to be the next revolutionary installment of the series, it should have a similar influence on the timeline as Ocarina of Time did, instead of being just another entry, where it would be overshadowed by the games around it, because it needs to fit their realities. And I personally like the idea of fixing the "damage".