I was having the same issue on my next Gen console. Wired controller, Wired internet, never had this happen on other games. All guns I tried using had a delay of 1 to 2 seconds before operating. I knew the game was 100% broken when I laid down scoped in, and every time I pulled the trigger it would wait a full second before shooting, often resulting in an incomplete kill, or my death.
The games not yet playable. Still a couple years off.
I had to completely reinstall the game for it to resolve itself. I think it has something to do with the sprint feature you can turn on and off in the settings. The feature is when your character lifts the gun up in the air when running. If you turn that to off which makes it only occur when you double press shift, he doesn't have a delay in firing.
Its called traversal sprint
