Traditional emergency alerts would take over all live media in the area, ie interrupt TV and radio broadcasts. In high risk towns like in tornado country they have literal klaxons that blare the warning to anyone within earshot.

As much flak as it got, the UK emergency alert system is a pretty good solution - collaborate with cell networks and the developers of the major phone OS's to push a notification to anyone in the at-risk area. It'll never truly be universal, but 80% of phones were capable of receiving the last test, and of those 7% didn't receive it. Honestly that's a much better reach that we ever would have had with radio and TV alerts.