It's been pretty much explained in this episode that the H-bomb/Incident caused the divergence of realities. So, Eloise shot Daniel and got his journal, learned about the "future", except in this timeline the Island sank. There's no need to force Daniel to be a mathematician, so she encourages his piano skills. The book told them that Desmond Hume was special, that he would be Daniel's Constant should anything go wrong. That's probably why Widmore gave him a perfect life, but without any attachments. Should he be needed, he'll have nothing holding him to this world but his service to Charles.

I don't think Eloise is connected to both sides like Desmond, Daniel or Charlie. She's connected because of the journal. This theory is by no means for certain, however, there is some logic behind it. Later episodes may offer further explanation into this topic.