On Android, FCM notifications sent with priority HIGH are usually available within 30 seconds and bypasses Doze (unless you're on ultra battery saver mode). Notifications sent with normal priority run within doze windows, so every 15 minutes or more.

Unused apps also indeed go in different App Standby buckets, which while it shouldn't affect FCM notifications, it does on some older android versions (https://developer.android.com/topic/performance/power/power-...)

If you're not using FCM, well you're limited to having to check for notifications yourself, either within doze windows, or by registering a foreground service that keeps your app alive and checking.