Watches and Warning should receive push notifications by default.
Advisories don't have push notifications turned on by default, so you will only see those when you open the app.
If you want to activate push notifications for advisories (which also covers special weather statements), tap the menu and select the edit location (pencil) icon then tap a location you wish to add advisory push alerts to, and then check the Advisory checkbox.
If you are seeing this on Watches or Warnings you may wish to check and see if the app is being optimized in your device's power settings. Go into your device settings to applications, to 1Weather, and then tap "Battery" are any of the power saving options on in that location?
If you go to settings --> Battery is 1Weather optimized under App Power Monitor? Is 1Weather listed in Unmonitored apps? If not, try moving it into Unmonitored apps and see if that helps.
If you go to settings --> Battery is 1Weather optimized under App Power Monitor? Is 1Weather listed in Unmonitored apps? If not, try moving it into Unmonitored apps and see if that helps.