Expect A Midnight Release
We'd expect the Bespin DLC release time to fall in line with the Outer Rim release time.
On June 21st, expect the Bespin DLC to unlock at midnight Eastern, 9PM Pacific in the United States. This is standard for most DLC content on Xbox One, PS4 and PC and we'd be surprised if Bespin strayed from the norm.
That said, don't expect a Bespin pre-load for Xbox One or PlayStation 4. EA DICE and its partners didn't offer one for Season Pass holders before the Outer Rim release date.
If you want to play ASAP, plan to be up late on release day.