No one has my theory yet? Awesome! This is going to be a wall of text, so here's my thesis/TL;DR right up front:
TL;DR: The Downfall Timeline, between Ocarina of Time and A Link to the Past. In other words, I think this Link is the same Link, the Hero of Time, that was defeated by Ganon.
My reasoning is as follows. I have no idea how I'm going to organize this and plan this out, so I'm just going to start typing.
My theory first sprang to mind when one of the first things Link sees is the Temple of Time. We already know that there are two possible Temple of Time's that this could be: either the one from OoT or the one from Twilight Princess. I feel like it's pretty fair to say with almost absolute certainty that--unless this is a completely new Temple of Time--this is the same one from OoT. The architecture is dead on inside and out. The geographical positioning in the world is a significantly better match for OoT than it is for TP. The developers were even kind enough so that our first sight of it was from the exact same camera angle we see it from when we walked down that back alley in Castle Town way back in 1998. The fact that its one of the very first things we see upon waking up likely denotes that it holds significant importance within Breath of the Wild's plot. I feel like the developers were not so subtly trying to bring OoT to the forefront of our minds with all of this Temple of TimeTM product placement here.
A lot of my remaining thoughts revolve around Ganon, so I think it will probably be best to continue of from there. It's been officially stated now that Ganon is once again returns as the main antagonist in this upcoming game in the series. Zelda lore is clear that all of the Ganon/Ganondorfs that appear in the series timeline are all the same human: The Gerudo thief that originally appeared in OoT. So this sets a clear marker, at least on the left-hand side of the timeline, that this game takes place at least after OoT.
Before I move on much further, I want to specify the distinction between Ganon and Ganondorf, since this has always been a topic of confusion within Zelda lore and fanbase. Whenever I refer to Ganondorf, I am referring to the human form of the Gerudo thief, and by Ganon I mean the beast form, typically represented as a wild pig or boar, that is summoned as the manifestation of the power of the Triforce of Power and Ganon's wrath. For further clarification, the Wikipedia article on Ganondorf is pretty good in its first few lines.
Now that a distinction's been made on the differences between Ganondorf and Ganon, all that's left to do is pick a timeline branch! This is luckily a really easy process of elimination because the King of Evil has consistently been referred to by a specific name in each branch of the timeline. In the Adult Timeline (beginning with Wind Waker) and the Child Timeline (beginning with Majora's Mask and TP), he has consistently been referred to as Ganondorf and appeared in humanoid form (save for a single battle in TP when he transformed into a giant boar and was temporarily referred to as Dark Beast, Ganon). All this leaves is the Downfall Timeline, where he is only ever referred to as Ganon (save for one maiden in A Link to the Past who, upon Link saving her, refers to him by his old name of Ganondorf as a form of defiance). In what we saw of BotW, Ganon was only ever referred to as Ganon, making this timeline a solid slam dunk.
All that's left is to figure out where in the Downfall Timeline this game takes place. Since Ganon has a nasty habit of constantly being killed/sealed away and resurrected in this timeline, this part could actually prove difficult. But I'll list out the evidence I have so far to give us some pointers in the right direction:
(1) Hyrule is unpopulated: Besides the old man and the Sheikah sages who had seemingly been sealed away underground for an exceptionally long time, Link didn't see a single soul on his journey through the demo playtime. (Caveat: There was the house structure that one of the player's raided that supposedly belonged to a farmer, but we never actually saw the character.) In addition, although Rupees are indeed a gameplay element, they don't appear to hold significant importance. The only Rupee counter was in the menu, not on the gameplay screen, and none of the players collected a single one in the multiple hours of gameplay. This would point to placing the game somewhere on the timeline when Hyrule is uncivilized, but not necessarily devoid of people. My top picks for placement based on this point is (1) between OoT and ALttP, because Ganon's rule caused the Hylians to flee even during the game, (2) between A Link Between Worlds and The Legend of Zelda, because although it might have been a limitation of the system at the time, there weren't any people besides the cave sages in TLoZ, and this game may be an attempt to explain why (Caveat: Zelda II had populated towns, but they were north of Death Mountain, beyond the scope of the original game.), or (3) after Zelda II, because this is still a possibility although we won't have much to go off of if this is the case.
(2) The huge emphasis on the Sheikah: Sheikah played their largest role in OoT, although even in this game they had supposedly died out already. This leans towards placing the game closer to OoT, if not before. This would however contradict the evidence of the ruined Temple of Time. In addition, if they are died out, the large amount of technology they have lying around after OoT is exceptionally confusing. If someone wants to try to explain either of these points, I'd love to hear theories, but I currently have more evidence to support my "after"-placement.
(3) Link already knows how to fight/navigate/adventure extremely well: This is surprising. From the moment Link wakes up, he's already climbing, scavenging, sneaking, kicking butt, and taking names like a seasoned adventurer. No tutorial, no mentor or teacher like in previous games. I'll just leave this point to sit and simmer because it ties into my next point.
(4) Link's lack of a Hero's Tunic: The next surprising point is probably the biggest upheaval of a Zelda convention to-date, and I doubt they'll just do away with it without some heavy plot point at hand. Link--a seasoned adventurer without a tunic. Why? My theory is that with his defeat at the hand of Ganon and the loss of his Triforce of Courage, he was disgraced of his title of Hero of Time, but was somehow saved from death and sealed away for a second time as Ganon used the Triforce's power to seize Hyrule. For how long he was sealed away remains a mystery.
(5) Other minor points:
Either the Bridge of Eldin/Lanayru is seen during gameplay. This would make since if in the Downfall Timeline after OoT since the game would likely be placed parallel in time to TP.
The presence of a Korok is another OoT time parallel to Wind Waker.
Although Sheikah technology is largely present, the Sheikah still seem to be dead. (Those sages practically looked like corpses held together by whatever magic contained them, and they disappeared into some magic dust whenever their duty was fulfilled. Regardless of if they were truly alive or not, they didn't look healthy enough for their people to be a currently-prospering civilization.)
Well, this took me well over an hour, but there it is. I really like the theory, as it offers the Hero of Time a chance at redemption, and the idea of Link dying in OoT making an entire timeline in the series always seemed like an anomaly to me for Nintendo, as they could have ended up doing this for practically any of the games in the series. Now that I think about it though, they have always used the phrase "the hero is defeated" when talking about it, never explicitly stating his death. Anyway, let me know what you think!