It was never believed that it was for real. Individuals that got the warning via text message may have believed it real, but no one in charge thought it was real.
The commander of the Hawaii National Guard had confirmation from U.S. Pacific Command that the warning was a false alarm within 3 minutes of the alert going out (per CNN). So, there was a maximum of 3 minutes between the text going out and the executive branch knowing it was a false alarm. Probably less, if you assume the commander of the Hawaii National Guard was not in the room when USPACOM confirmed it was false alarm.
Any ballistic missile launched will be detected within seconds by U.S. satellites and/or other assets and the Pentagon will know immediately and the appropriate players will be in a meeting. Every potentially relevant combatant command has a 24/7 watch floor that is always staffed, along with the Pentagon and White House Situation Room (and likely others).
Also, North Korea cannot secretly launch a ballistic missile… it takes a lot of prep work and we know when they get to the point that “they can launch at any time” so if none of their missiles were in this ready state, anyone in the know would assume the alert is wrong, and then immediately make the appropriate phone calls to double check and make sure no one else was seeing something different (via satellite, etc).
I have no info on the political side of things, but it is likely that “Hawaii accidentally tells its residents that a missile is coming at them” does not warrant immediate notification of the President. And before someone tells the President that a missile is coming at the U.S. (which would definitely warrant immediate notification), they would’ve needed more than a random text message.