She gets better. Nu gets Lambda's soul merged with her own AND Lambda's memories as well. It helps her fight off Terumi's mindfuckery.
She survives Chronophantasma and is now on Ragna/Celica's side. Sorta.
Wrong.
Lambda was pretty much soulless that Nu's soul was implemented into her body (Hence why Kokonoe said when Lambda was suffering, she told her that's not her memories). Afterwards she just becomes a mindless drone until she regains her memories about Ragna and such, then saves him from getting killed by Terumi. Lambda's body perished after giving her Idea Engine to Ragna, and Nu's soul went back to her own body pretty much in the Cauldron (since she was regenerating there).
As for Chronophantasma, she gets reprogrammed/tuned again by Terumi/Hazama and becomes him and Relius' pawn again.
I guess you didn't see what happens well. Ragna and Celica did indeed take Nu out of Takemikazuchi, after Ragna defeats her. However Izanami/Saya appears, making Nu stab Ragna in the back, then making Ragna go berserk in Black Beast mode, where he wipes the floor with Mu-12 and Jin, then disappears. It was not even shown what happens to Nu next after she backstabbed Ragna.
Nu's personality from Continuum Shift pretty much disappeared during Chronophantasma. She went back to her Yandere personality, except the worst thing was how she was being used by Terumi and Relius. Some theorize it's because of Nu's Blazblue and Terumi/Hazama's programming towards her.