Madoka and Takato’s routes belong to the Black Bunny route. Nakaba’s route is actually the borderline between White and Black Bunny, but since you need to clear Madoka’s route first to get to Nakaba, here’s the Black Bunny common route.
It’s pretty similar to White Bunny except instead of waking up at Yuushinkai headquarters, Nadeshiko wakes up in the government building and is greeted by Rain, Madoka, and Takato.
When Nadeshiko wakes up in the Distorted World, she finds herself at the government building, also known as Clock Zero, which started out as research facility. There, Rain reveals that everything Kaga Akira did was for the sake of “bringing her back to life.”
The reason why the world is destroyed in 2020 is because King (aka Kaga Akira, Kaidou Takato) destroyed it. After she got into her car accident in 2011, he threw himself into his research work. He was still a middle school kid at that point, so he asked his dad to build a research facility as one of Kaidou Group’s enterprises.
Takato had been researching ways to cure her when his research ended up in an explosion in 2016 and caused the world to be what it is now. He was surprised and sad but glad at the same time because, thanks to the explosion, he finally completed his research and was able to figure out a way to bring her back to life. Since then, he founded the current government and acted as “King,” in order to create his ideal world: a world where no one gets hurt.
To realize his ideal world, the government imposed strict rules on the people, which includes branding them with “identification code,” which allows the government to know where they are at all times, in addition to giving them monitored work schedules and enforcing curfews. Only the King’s trusted government members are given titles and able to walk around without an identification code. Rain (“Rook”) and Madoka (“Bishop”) are included in this category.