Dick's Story
For starters, there's a law accounting for children in this world that offers them protection, rights, and services for being born with or developing powers and abilities. This covers metahumans, mutants, and child vigilantes.
In terms of Dick, the Court of Owls will be much more prevalent in his life, being the main reason he becomes a vigilante. The Court of Owls will also have a much larger reach in Gotham and the world, with Haly's Circus being connected to the recruiting process. Infants and toddlers are monitored to see who would be talons due to their legacy and potential. Dick was chosen to be a Talon when he was maybe about two years old. He was also secretly conditioned and trained to be a talon as a child, with the plan being to take Dick when he was ten years old to formally begin the process. They secretly experiment on him to prepare him for being turned into a Talon, putting Dick under the law mentioned above. This plan is ruined when Mary and John are killed by Tony Zucco when Dick is eight years old. They decide to expedite the process by making arrangements for Dick to be taken sooner. Because of this, they get Dick to stay in Gotham and have him placed in Juvie to take him through a plant in Social Services. Again, their plans are ruined, this time by Bruce Wayne getting to Dick before they could.
Dick will still want revenge on Zucco, and he decides to go after him secretly at night. He does decide not to kill him due to Bruce's influence, but more indirectly with Bruce talking to Dick and it getting through to him, not realizing that Dick wanted to kill Zucco. His fighting style at this point is purely acrobatic, but Dick will also have a taser to subdue criminals. Once he brings Zucco to justice, he finds himself wanting to prevent similar tragedies from happening to others. Thus, Robin is born. Robin becomes a symbol of hope, as well as a bit of an urban legend for a while. Optionally, some of the really dangerous Gotham villains can not exist since Batman won't exist, particularly the Joker and Bane. The villains can also be weirdly protective of Robin, later Nghtwing, at times.
In jail, Zucco is murdered by the Court of Owls for his transgressions and for costing them Dick. They make it look like a suicide.
Dick can be home-schooled at first since he's still working on speaking English. He can also take self-defense classes and martial arts classes after suggesting it to Bruce, who thinks it's a good idea. Bruce can know some fighting, as well. This can lead to them bonding over sparring sessions, which are much safer than canon. Dick can also develop his detective skills, which can also be a hobby or talent of Bruce's still. Other skills can still include stealth, spying, undercover work, and technology. His weapons can be more homemade, like Peter Parker as Spider-Man.
By his preteens, Dick officially learns about the Court of Owls. As a teenager, maybe around fifteen, Dick make arrangements to keep Bruce save by pretending to be in a program in California. At this time, he'll join and lead the Teen Titans for a few years. Deathstroke develops his interest in Dick around this time after he goes undercover as Renegade. Once he's eighteen or nineteen, Dick will transition into Nightwing after hearing the story from Superman.
Dick keeps his identity super secret to protect Bruce, as well as his siblings. Basically, no one knows who Robin or Nightwing is.
The Court of Owls will be his main enemy. They will constantly plot to get the Gray Son in their service. One of their plots leads to them making a deal with the League of Assassins, which is the reason Talia was at the party and ended up meeting and falling in love with Bruce. She becomes an ally of Dick's until she dies. Once she dies and Damian gets to Gotham, Dick wants to protect him. After he arrives, Bruce can get kidnapped, probably by the Court of Owls or an associate, instead of being trapped in the time stream. Dick takes Damian under his wing to protect him from Ra's Al Ghul, and they develop their bond during the year that they are looking for Bruce, who is eventually rescued.