So first,

Let’s discuss the reason I believe she left the Fatui, which I theorised here before it was confirmed.

As we can see in image 1, her statue nearest to the Fatui Base is caged. In image 2, from the web event, they suggest it is to harness its power, and showcase their dominance.
Trying to harness its power could be because of the reason that Columbina left the Fatui. This could be because they wanted to have access to Lunarium Force, which, Columbina disagreed with. This could’ve sent her into conflict with the Fatui.

Now that’s pretty much all for my theory on why she left, I think it’s because they wanted the moon power and she didn’t want them to have it, and wanted to protect it. 

Now as for how it affects the Fatui,
In the Gods’ Limits trailer, we heard that Pierro is looking at the factors and consequences of killing a god. Now, Columbina’s power could be necessary, and i’m not trying to say ‘Columbina is the most powerful Fatui!!’ But, she has access to a force none of the other Fatui members hold. She Is also one of the most powerful fatui, meaning, the fatui missing her could affect their plan heavily. The rise of a new moon, while the moon goddess of said moon had ties with the Fatui, could’ve also been a way for the Fatui to get outside of the firmament, which is where Celestia is. 

Now for the last part,
How long will her separation last?

Now from what I took from arlecchino’s dialogues, you don’t get to leave the fatui, You can try seperate from them, but joining them is permanent. Now as we heard in the Gods’ Limits trailer, they actively want to bring Columbina back, infact, that’s why the entirety of the Fatui were sent to Nod Krai. (image 5)
As image 3 shows, they used the time for her departure as ‘Right now’ which, suggests it won’t last for long. ‘Right now, for uncertain reasons, The damselette has left the Fatui.’
The wording used suggests the departure won’t last permanently. The next picture on (image 4) Also shows that she is hiding in silver moon hall. Referring back to image 2, they also have her statue in a cage to show dominance. She is hiding because she is aware she can’t beat the whole Fatui, and, they could take her if they found her, and, the statue being caged is not only to try gain its power, but to warn that they will keep her trapped, no matter what.

Why Columbina left the Fatui, how that affects the Fatui, and whether her choice is permanent