The AEA app is separate from the cellular broadcast technology your provider uses to send wireless alerts. You will need to contact your cellular provider to ask why you did not receive the test alert.
In the meantime, the AEA app provides a fallback option. You should receive the alert in the app as long as you have a network connection, and are not offline or in airplane mode. In addition, there is an Override Silent Mode setting so that you always hear critical alerts even if your phone volume is off or in do not disturb mode.