Try running checklist below - To make sure settings are correct:

1)Long press on Gmail icon to reach its app info page. Allow all notification from it.

2)Launch the Gmail app . In the top left, tap Menu . Tap Settings. Select your account. Tap Notifications and select a notification level.

3)Long press on Gmail icon. Find Battery > Optimize battery usage > Change the setting to Unrestricted

4)Turn on: Settings> Accounts> Automatically sync app data(At the bottom of screen)

Further reading  

Try running the checklist below - To make sure the settings are correct:

1. Long press on the Gmail icon to reach its app info page. Allow all notifications from it.

2. Launch the Gmail app. In the top left, tap Menu. Tap Settings. Select your account. Tap Notifications and select a notification level.

3. Long press on the Gmail icon. Find Battery > Optimize battery usage > Change the setting to Unrestricted

4. Turn on: Settings > Accounts > Automatically sync app data (at the bottom of the screen)

Further reading