WARNING: EARLY CHAPTER WARD SPOILERS AHEAD, DO NOT READ IF YOU DO NOT WANT MINOR WARD SPOILERS
I don't actually know how to do the spoiler thingy plz no gib bannu
I would like to refer people here to Damsel of Distress and, by extension, what we know about cape clones.
Powers, thinker or otherwise, are black boxes for data. They tend to record information, particularly on conflict of capes. As a result of this, Despite damsel of distress being a half assed clone only meant to have a "close enough" set of falsified memories and impressions to look something like the original, it actually managed to develop distinct memories from Damsels Career. These memories tended to focus primarily on cape life, fights, and events post-trigger, but included vaguer but still present memories pre-trigger as well.
Assuming tattletales massive brain trauma is healed (one way or the other), its likely her shard will passively fill in the blanks by copying over its own blackbox data.
This is ignoring that tattletales power is uniquely suited to deal with amnesia, it allowed her to see through imps power, something that induced amnesia in EVERYONE around her, and it can presumably be used to allow her to turn vague memory impressions into full ones or even to possibly remember phrases shes forgetting. As is it seems a lot of her amnesia may be some mix of massive brain damage and a disassociative fugue state she is currently in.
The latter isn't like your breaking bad fugue state, its a thing generally found in victims of traumatic events or of people with mental disorders causing it where they'll suddenly forget who they are or delude themselves into believing they are someone else entirely, taking on a differentish name and not actually recalling much of who they were. It will come back to them over time and much of it'll come back if prompted, but its a known phenomena and in this case I suspect the trauma of being burned alive, dropped off the rig, and generally turned into a cyborg is currently fucking with her almost as much as the brain trauma is. Probably more if her shard is just waiting to fill in the gaps.
EDIT: Reading back on the post, tattletales shard explicitly lists "Psychologic dissassociation, psychogenic amnesia: exacerbated by physical trauma."
Tattletale isn't actually coping well with not being in the same body, in fact I'd go as far as to say that straw, coupled with having her brother kill herself, a gun held upto her head, the lung fiasco, life under threat, being a villain due to force, and getting rekt in a bank robbery has all led to her deciding to pack her bags and go home for now. Her memories are all still there, shes just actively refusing to access them on some subconsious level.
Given time, Therapy, good living (lol like this'll happen), and some level of improvements and she'll PROBABLY go back to mostly being herself, although she may need mechanical assistance with the brain damage she seems to have to several bits of her cognition ability, that said her shard may help there.