Placing this game on the timeline is very hard xD
Lets go through each timeline and see how it could fit.
Adult timeline
So, the most obvious connections are the inclusions of both the Koroks and the Rito.
We know from Wind Waker that the Kokiri took on the forms of the Koroks after The Great Flood, to be able to fly to other islands and plant deku seeds to "one day unify all the islands into one"
The Zora evolved into the Rito also due to The Great Flood, but this was influenced by the Godesses so that they didn't find hyrule castle under the waves.
In BotW, there is an item called Rock Salt which is found all over Hyrule. Its in-game description says the following:
Crystallized salt from the ancient sea commonly used to season meals. Cannot be eaten in this form.
The Ancient Sea is most probably refering to the Great Sea, though some argue that its the Ancient Sand Sea from Skyward Sword, which its imposible since the rock if found even on the tallest of mountains.
Also, in the Japanese and German versions of the games, they mention the hero who sails the seas. This are the extracts (She s speaking to the master sword, in Memory #1):
Japanese: "You cross the seas when you seek the gold made by the gods. May you be alongside the hero."
German: "Whether the hero crosses the sea, or creates a link to the past, may you always be at his side." (Note how she mentions A Link to the Past)
Now, the last time we saw the Master Sword in the Adult Timeline it was plunged in Ganondorf's skull, in ganon's tower. The Master Sword in BotW is in the Korok Forest, protected by the Deku Tree.
Also, New Hyrule was stablished, if the sea was drained at some point, maybe by the actions of the Koroks (which is unlikely since they are acting against the very will of the gods themselves, as King Daphness Nohansen Hyrule wished for hyrule to be sunken forever and the gods granted his wish), but lets say the Koroks manage to drain the sea and they discover Old Hyrule. Would New Hyrule merge with Old Hyrule or would both the Kingdoms coexsist?
Although BotW has many connections to Wind Waker, I find it very unlikely that it will be placed in this timeline.
Child Timeline
Lets start with the most obvious connection. Memory #1, Zelda mentions Skyward Sword, Ocarina of Time and Twilight Princess. "Whether skyward bound, adrift in time, or steeped in the glowing embers of twilight...The sacred blade is forever bound to the soul of the Hero."
Arbiters grounds is in Gerudo Desert, although just merely ruins of what it used to be.
In Twilight Princess HD, there is stone carvings in the walls of hyrule castle that depict the Gorons, Zora, Rito and Oocca. (Note that this was added only in the HD version, and Aonuma said that there were various things in the remake that when you played BotW you would have "Oh, I see" moments)
But there are not only references to twilight princess, but also to Majora's Mask, such as lakes and locations named after characters from the game.
As for the Sages. Yes, it is said Ruto was a sage in BotW, but Ruto could still have awakened as a sage after Link was sent to the past. Also, Link might have returned from Termina after Majora's Mask, given the Hero's bow to the Gorons, and joined the Royal Guards. Link could have died in battle and burried under that big tree in hyrule castle, where the ghosts surround it. But this is just speculation.
Downfall Timeline
The main inspiration for BotW was the Original legend of Zelda, and even the Tunic of the Wild is based on the Tunic of the hero of that game.
The Master Sword resting in the lost woods is a clear reference to Alttp, and even the placement of the Silent Princeses to the flowers in Alttp are spot on.
There are locations named after places or even straight from this timeline, such as Tal Tal Peak, and twin peaks.
Aonuma stated that Hyrule has faced Ganon for ages, and Impa says this in-game. The downfall timeline is the one that has the most confrontations with Ganon (and yes, Ganon, not ganondorf)
The triforce is not mentioned in the game, well, it is in the Japanese version as stated earlier. HOWEVER, if you look closely at zelda's hand in the last memory when she uses her divine power, she has all the pieces of the triforce. All the triangles glow, and if she only had the the triforce of Wisdom, only one of them would have glown, as seen in MANY other zelda games. And Link is not once seen with the triforce mark on his hand. And Ganon doesnt have the triforce piece either.
So, it is clear that Breath of the Wild has connections to every game in the franchise. Nintendo has everything thought out, and they are certainly not going to screw the timeline over. We just have to wait and get the confirmation.
Also, just something cool I, and many other have found while playing BotW.
In Gerudo Desert, Eldin and Hebra there are Leviathans Bones, which are part of a sidequest. They strongly resemble whales that we have seen befoee in other games, well, at least two of them.
In Eldin, the remains belong to Levias, the Sky Spirit from Skyward Sword. The flat surface ontop of its head and the blowhole give it away.
In Gerudo, the remains belong to The Wind Fish, the whale from Link's Awakening. This is clear due to the wing bones on its back, and the blue whale skull structure.
The one in Hebra is unlike anything we have seen. It looks like a whale, with an elongated skull. No whale deity known matches the bones. Maybe we will get introduced to it in the future.