All the notifications in Widows 10 are shown in action centre only (even for old win32 apps). If there is no notification there that means that chrome is not showing desktop notifications.
Make sure Google chrome is allowed to run in background.
- Go to Google chrome settings page. Make sure you have checked the option to allow Google chrome apps to run in background even when Chrome is closed.
- If you are using a laptop make sure battery saver is turned off.
If these things are set correct, Clear cookies from chrome and re-enable desktop notifications.
If you weren't asked to enable desktop notification by gmail, clear all data from chrome and try it again. The whole point is to turn that off and re enable the desktop notification. If this doesn't help you re-installing chrome may help but I am not sure. Also give the built in Mail app a try if other things fails.