I don't know how it works exactly, however, I do know that I usually get push notifications, or they show up on the web/app before the delay is known to GA's or shows up on the screens at the airport. Obviously, its got to flow through different systems to get to the different places.

One agent, a couple of years ago, told me they notify the flight crew before posting the delay on airport screens - don't know if that is accurate or not.