Ok I know that this a long shot but if possible heres all the things I remember about the story
•Its been a very long time since I've read it possibly 10 years or so
•The OC (I can't remember her name) turns out to be a time lord child
•It was a Rose bashing fic, and Rose very much did not like the OC
•The Doctor ends up leaving Rose behind to better focus on taking care of the OC
•It had School Reunion, in which she accidentally starts cracking the the Skasis Paradigim when the Doctor is talking to Mr. Finch
•Also in School Reunion I believe Rose finds out the Doctor had in fact left her behind and wasn't planning on coming back for her
• According to the above ^ The OC felt really guilty for leaving Rose behind and managed to convince the Doctor to pick her up and travel with her again
•She is time sensitive and can see the timelines
•The fic also contained The girl in the fireplace and I know during that episode for some reason the OC ends up on the med-bay in the TARDIS and she's looking at the timelines when the Doctor get stuck in France (after crashing through the mirror window near the end of the episode) and Rose yanks a saline drip out of the OC because she wasn't responding (as she had been looking at the timelines) and Rose thought it was something keeping her asleep and when the Doctor finds out about this he becomes very upset