In the latest version of the emergency broadcast specification (WEA 3.0), a smart phone’s GPS capabilities (and other location features) may be used to provide “enhanced geotargeting” so precise boundaries can be set for local alerts. However, the system is backwards compatible – if you do not have GPS, you will still receive an alert, but whether it is displayed depends on the accuracy of the location features that are enabled. If the phone determines it is within the target boundary, the alert will be displayed. If the phone determines it is not within the boundary, it will be stored and may be displayed later if you enter the boundary.
If the phone is unable to geolocate itself, the emergency message will be displayed regardless. (Better to display the alert unnecessarily than to not display it at all).
The relevant technical standard is WEA. Only the latest WEA 3.0 standard uses phone-based geolocation. Older versions just broadcast from cell towers within the region, and all phones that are connected to the towers will receive and display the alerts. You can read about it in more detail here.