After updating your iOS device to iOS 8, Hangouts app’s notifications stop working. Below are some possible solutions mentioned in Hangouts Help Forum:
Workaround: After upgrading to iOS 8, please make sure you open the Hangouts app at least once.
Workaround to turn notifications on:
Step 1: Touch Settings on your iOS device.
Step 2: Touch Notification Center.
Step 3: Touch Hangouts.
Step 4: Touch the toggle button next to “Show on lock screen” so it switches to green. The notifications will appear even when your device is locked.
Step 5: Choose what type of alert you’d like:
Banners: This shows you only the text from the notification.
Alerts: This shows the text from the notification and gives you the option to open the app.
Step 6: Touch the Home button on your device to go back to the main screen.
If you still facing the same problem after follow above workaround, you may like to post your comment at Hangouts Help Forum