Canon Timeline
The younger version of Ziti that is born after the Hamatos take care of the Krang threat is still mutated by Big Mama. Since her properties and resources weren't destroyed, Ziti is able to be raised by Big Mama's secretary, Ria. Considering her young age and softshell, Ria was beginning to look for ways to get Ziti out of Big Mama's clutches, because the Battle Nexus would tear that little girl to shreds. Ria then proceeds to drop off a toddler Ziti in the abandoned subway station the turtles made their home, and flees back to the Hidden City, claiming the softshell ran away. Cass and April, freshly married with a young Junior already born, are quick to take Ziti in as one of their own.
Future Timeline
In the timeline where the Krang invasion is successful, Ziti is found and taken in at a much younger age, found during a supply run in the Hidden City. She's trained to defend herself at a young age by Raph, who alongside Cass helped to train all of the younger children to be able to hold their own in case they ever find themselves in danger. Ziti proudly takes on the role of older sister to both Junior and Fusilli (though she did take a little while to warm up to the young turtle) and is highly protective of her family. Throughout the years of the apocalypse, she grows bitter and quiet, as the shrine of their fallen family members grows greater in size each passing week. By the time the ROTTMNT movie is set to start, she is bitter and not interested in helping save this other timeline's Earth, when they were forced to abandon their own timeline. She finds herself increasingly disappointed in this world, feeling everyone is ungrateful for the stability in their lives. Most of that bitterness is taken out either on herself, of current timeline Leo, because he's nothing like the father and sensei she remembers.