Gotham Knights is an upcoming television series set in the wake of Bruce Wayne's death. It tells the story of Turner Hayes, Bruce's adopted son who has been framed for his murder.
What does this have to do with Gotham? Danny Cannon is directing the pilot, for one. He is a major creative force in the Gothamverse, consisting of Gotham and Pennyworth. Two of the writers for the show, Chad Fiveash and James Patrick Stoteraux, were the co-executive producers of Gotham S5.
While all three writers worked heavily on Batwoman as well, this show is said to not be a spin-off of that series, so for the sake of argument I'm going to discount that at the moment. One thing I will note is that Renee Montoya from Gotham did return on Batwoman, so they aren't opposed to bringing elements of Gotham onto their Batman shows.
As for how the show connects?
Gotham Knights, despite filming in a different location to Gotham, uses the same city skylines. Regardless of what reason they did this, it was no mistake. Additionally, they added two buildings that weren't seen in the Gotham skylines. Wayne Tower and a tall glass building that seems to be the same size if not taller. In the Gotham finale, Barbara Kean had a tall glass building under construction that she said would be taller than Wayne Tower in three months. They also removed a building in the far right possibly to keep continuity that Barbara's building is the tallest. The only other show to reuse skylines from Gotham is Pennyworth, which is set in the same universe. The blimps in the sky are reminiscent of Pennyworth.
Wayne Manor in Gotham Knights has the same fictional address it does in the Gotham series. This fictional address is taken from Batman: The Animated Series and is a reference to the episode "The Demon's Quest". If it is the same continuity, Gotham Knights' version would be the rebuilt Wayne Manor after Jeremiah destroyed it during No Man's Land. While Alfred obtained the original blueprints to rebuild it, it can be assumed that it was changed or expanded upon.
To support that idea, the gate to Wayne Manor is identical to the version seen in Gotham.
The GCPD logo in Gotham Knights appears to be an amalgamation of the Gothamverse and Nolanverse logos, leaning closer to the Gotham design.
Here's an exciting prospect. There is a man at Bruce Wayne's funeral who seems to resemble Sean Pertwee's Alfred. The hair, the nose, the ears, everything is pretty similar, at least from this angle. It's possible that it is a funeral pastor or priest, because he is carrying a cross necklace. However, Alfred is a Catholic in the comics, and pastors/priests usually carry a model cross or the bible rather than a necklace. Based on the fact that the writers previously brought back Renee Montoya for Batwoman, it wouldn't be totally surprising if they did decide to bring back other actors from the Gotham TV series to reprise their roles. This is speculation however.
Naturally, this theory would make Harvey Dent in Gotham Knights a continuation of the version seen in Gotham S1-2. Nicholas D'Agosto aged into Misha Collins over the past 30 years between the shows. A teaser for Gotham Knights tells us that Harvey's age during the series is 56. Misha Collins is 48 for reference. This would make Harvey in his 20s during Gotham. One issue that presents itself, however, is that the Pennyworth tie-in comic references Two-Face while Bruce is alive. My theory is that, after the Gotham Finale, Harvey Dent is scarred and becomes Two-Face. Alfred recounts a Two-Face in the Pennyworth comic book. Prior to Gotham Knights, Harvey Dent uses prosthetics or surgery to cover it up and represses his mental illness. Maybe people knew Harvey was Two-Face, maybe they didn't. Regardless, Harvey continues his job as District Attorney. We're likely to see him relapse in Gotham Knights.
Of course, we know the Court of Owls is involved in the series, and we know they were "destroyed" in Gotham S3. However, all it would take is the next in line to take over leadership or one figure to rebuild the organisation. We know that Lincoln March is involved in the show. Lincoln March in the comics is Bruce Wayne's supposed older brother, the first-born son of Thomas and Martha, Thomas Wayne Jr. This character actually has ties with the Court of Owls, and is potentially a major villain in the upcoming show. What If Lincoln March, instead of being Bruce's brother, is his clone, 514A from Gotham? 514A was even created by the Court of Owls, so it would make sense if he became Talon.
Another possibility is that March is a misdirect. What if the real main villain was actually Samantha Wayne, Thomas and Martha's daughter from Pennyworth? We have no idea what happens to her after the events of Pennyworth.
We'll just have to wait and see what connections or contradictions present themselves when Gotham Knights finally releases.