Batman would be heartbroken. He has trained Robin and sees him as a partner and almost like a son. He would first try to talk to Robin, to find out what made him turn evil. He would hope that he could reason with Robin and bring him back to the good side.
Another plot could be that he is exposed to some sort of mind - controlling substance or technology. The enemy could be using it to turn him against Gotham. Under this influence, he starts to see his friends as foes. He could begin to attack the Bat - family and other superhero allies. His actions would be erratic and violent, and the city would be in chaos as they try to figure out what happened to their hero and how to stop him.
The Joker would probably be both delighted and a bit confused. He's used to Batman being his nemesis, but an evil Batman would change the dynamic. He might initially see it as a new and exciting game, but then realize that an evil Batman could be a real threat to his own chaotic existence. The Joker might try to manipulate the situation to his advantage, but it could easily backfire on him.
One possible plot twist could be that Robin is influenced by a powerful and evil villain. Maybe the villain uses some kind of mind - control device or magic on Robin. For example, the Joker could find a way to make Robin see Batman as the enemy. Robin might then start attacking Batman and helping the Joker with his evil plans.
One idea could be that Batman is turned into a cat by a villain's new experimental device. As a cat, he has to rely on his natural feline abilities like stealth and agility to still protect Gotham. For example, he could use his cat form to get into small spaces in buildings to gather evidence or spy on criminals.
Another possibility is that Batman is experimenting with a new technology to enhance his abilities. He creates a device that is supposed to give him the ability to fly like a bat. However, something goes wrong during the testing phase. The device fuses with his DNA, and he starts to turn into a bat. He has to deal with the physical and mental changes, all while trying to keep it a secret from his allies like Alfred and Commissioner Gordon.
Well, he might find it really hard to operate his Bat - gadgets as a cat. His paws are not made for pushing buttons or typing on keyboards. So he has to figure out a way to adapt. Maybe he uses his cat claws to scratch out messages for Alfred to understand what he wants to do.
Perhaps it's due to a psychological breakdown. Batman has endured so much trauma over the years. The constant fight against crime, the loss of loved ones, and the never - ending battle with his inner demons could all take a toll. One day, he just snaps. His mind can no longer handle the pressure, and in an attempt to escape from his own tortured self, he subconsciously adopts the persona of the Joker, as the Joker seems to have no such moral and emotional burdens.
He would become more cynical. His once - positive view of the world and people would be replaced with suspicion and hatred. He might start to see everyone as an enemy.
One possibility could be that he's brainwashed. Maybe a powerful villain uses some mind - control device or magic to turn Robin against his friends. In fanfiction, this could lead to some really intense battles where his former teammates are reluctant to fight him at first, but realize they have no choice.