Do we know, with clues in the game, where Breath of the Wild happens in the Zelda Timeline?
Are there at least clues to find out if we are in the Fallen Hero, Zelda or Link timeline?
The timeline in which the story takes place as you see fit. Based on the other answers, many people have different interpretations on which timeline Breath of the Wild takes place.
A new article from IGN contains information from an interview with the director of Breath of the Wild. Stated in this interview, the timeline is which this story takes place is based on the players own imagination and how they interpret it.
In an interview with Famitsu, (translated by Siliconera), Breath of the Wild director Hidemaro Fujibayashi says exactly which timeline it rounds out is "up to the player’s imagination."
The current answer was right to say how it contains elements from the other timelines. This made it is unclear exactly where and which timeline Breath of the Wild fits into. Based on the article, the Japanese website of Nintendo's Zelda timeline has been updated to contain Breath of the Wild. Oddly enough, Breath of the Wild is at the end and is disconnected from the other timelines.
The reason for this is because the history of Hyrule in this story went through many revisions, including elements that either fit perfectly or warranted a change.
"Hyrule’s history changes with time," explained series producer Eiji Aonuma. "When we think of the next game and what we want to do with it, we might think, 'Oh, this’ll fit well', and place it neatly into the timeline, but sometimes we think, 'Oh crap', and have to change the placement. Actually, the decided history has been tweaked many times."
There is no definite answer currently. At present, Nintendo has not commented on the placement in the timeline other than saying it is after Ocarina of Time. In the game, there are references to each timeline branch, which makes it quite confusing to sort out.
Others have already pointed out the ceremonial speech wherein Zelda makes references to Twilight Princess, Skyward Sword, and Ocarina of Time, but they've failed to point out some other important pieces.
First of all, the Divine Beasts are all named after the sages from Ocarina of Time (with the exception of the Rito divine beast). This is confirmed in the game's lore for Vah Rutah and Vah Naboris by a stone tablet in Zora's Domain and by Urbosa in a cutscene. Vah Medoh is presumably named after Medli and Vah Rudania after Darunia. But Medli only appears in the Adult Timeline, so the only time where we have all four of these sages is in this timeline.
Also on said stone tablet in Zora's Domain are these words:
It is said that Ruto then awoke as a sage, facing this foe alongside the princess of Hyrule and the hero of legend.
There's now a problem with this being in the Child Timeline. According to Hyrule Historia, the Hero's Shade in Twilight Princess is the Hero of Time from Ocarina of Time. He was never acknowledged as a hero and nothing in the lore indicates Ruto became a sage in the Child Timeline either.
Some weaker, but noteworthy evidence:
The Great Deku Tree is alive. In the Child Timeline, he is presumably still killed by Ganondorf's curse, but we have no confirmation that he was revived because we don't see him in Twilight Princess, Majora's Mask, or Four Swords: Adventures. In the Adult Timeline, we see him revived as a sprout (presumably this event also happened in the Downfall Timeline).
At Hyrule Castle:
In a secret room in the library at Hyrule Castle, you can find the King's private study. In his diary he mentions that he named Zelda according to the ancient tradition. The tradition of naming females born in the ancient line Zelda is only explicitly stated in the Downfall Timeline. This is stated in The Adventure of Link, and each princess is named Zelda after the princess who was afflicted by a sleeping spell. So this points to this game being in the Downfall Timeline.
At the end of the game, Zelda calls Link "The Hero of Hyrule", which is what happens at the end of the original Legend of Zelda. The beginning also mirrors the original game because you meet an old man in a hood who helps you.
The Resurrection Chamber: The Downfall Timeline is the only timeline where we know Link previously failed to defeat Ganon and died. The Resurrection Chamber might have been created with that in mind. This is also a thematic reason that the game could be in the Downfall Timeline (thematic, meaning it's another story of a hero failing). The Downfall Timeline also has the most appearances from Ganon, and past conflicts with Ganon are mentioned in the game.
The Rito and Korok: Early on before the game was even released, there was a great deal of speculation that this game would be in the Adult Timeline solely based on the existence of the Korok and Rito in game. Previously, these races have only been in the Adult Timeline.
So with this conflicting evidence and more, there can be no definite answer, only theories.
In memory sequence #1 Zelda is holding a ceremony to honour Link and says (paraphrased by memory)
be it skyward bound, through time or in to twilight…
This references Skyward Sword, Ocarina of Time and Twilight Princess, respectively. Thus, Breath of the Wild falls in to the middle timeline split, where the Hero of Time succeeds in defeating Ganondorf and returns as a child.
I believe it might be the Fallen Hero Timeline because the Master Sword is placed in the Lost Woods and it only shows Ganon in the game. It can't be the Child Timeline because the Master Sword is placed in the ruins of the Temple of Time. It can't be Adult Timeline either because it's not in new hyrule, Ganondorf was stoned by the Master Sword underwater,l
-- Warning: Minor spoilers --
Based on an IGN article (Timeline: BOTW), I believe Breath of the wild takes place on the Windwaker timeline, sometime after Windwaker. I have two theories about when it could be.
Edit: Link in BoTW would have to be the son of Link from OoT (from Malon(?)). This is due to Mipha mentioning how they grew up together, as this Link must have traveled all over Hyrule after OoT with his father. Zelda in BoTW says "Your path seems to mirror your father's", as in to protect the current princess. Notice how most of the Divine Beasts are named after sages from OoT, this was probably the Hero of Time/this Link's father's idea.