The film takes place at some point after Batman v Superman: Dawn of Justice, but before Justice League.
Early in the film Amanda Waller states that a special team needs to be created due to recent events involving Superman. She and others consider him a threat, and that other metahumans need to be formed into a team to protect the US and her interests.
Per dialog from the film:
Gentleman, ladies: what if Superman had decided to fly down, rip off
the roof of the White House, and grab the president right out of the
Oval Office. Who would'a stopped him?
Given he was at a congressional hearing that resulted in the deaths of everyone present except for him, leading people to believe he was the cause of the destruction, and coupled with his presence along with others such as Batman possibly causing an increased level of aggression in certain criminals (a theme explored many times before), it's only natural to want to get something in place to protect national interests from threats non-metahumans are likely to stand a chance against Superman and others like him. After all, he did go on to fight Doomsday not long after that hearing, a foe no one from the military was capable of countering, and taking Superman, Batman, and Wonder Woman to take down, supposedly killing Superman in the process, something no human had managed to do.
Batman v Superman: Dawn of Justice, as you note, shows that Bruce obtained video of other metahumans from Luthor and gave it to Diana, one of which depicts The Flash stopping a convenience store robbery. This serves to clearly establish that The Flash is at least somewhat active prior to the events of Batman v Superman. With Suicide Squad we not only see Batman helping to bring in some of the squad's members, but in a mid-credits scene we see him obtaining more exact information regarding the other metahumans he learned about in the videos he stole from Luthor. I'd argue this indicates that while he was aware of the other metas out there, he didn't know who they were or where to find them, for which he had to rely on Amanda Waller to provide that information in exchange for his protection.
While it's not directly related to the films so far, a few years ago with their The New 52 reboot of DC's comic books, the first comic released as part of that reboot was Justice League. The story begins with Batman and Green Lantern learning that rumors of the others' existence are actually true, which provides credence to the idea that even the heroes that eventually form the Justice League were unaware of one another at some point in time, only hearing rumors and stories being passed around.
As such I'd argue it's entirely possible for the superheroes that will eventually be involved in Justice League late next year also don't fully believe the stories of other metahumans being out there, and in fact in a recent Comic-Con preview of Justice League, there's this bit of dialog between Cyborg and Batman:
Cyborg: I heard about you. Didn't think you were real.
Batman: I'm real when it's useful.
I mean Batman has supposedly been active for some time, and Cyborg seemingly still thought he was just a myth until actually teaming up with him.
