Windows Mail App does not work with Gmail accounts

I suggest you open the mail app. Click on the Gmail account and right-click on the account and select account settings. You need to click on mail change mailbox sync settings and make sure that the email sync option is checked.

If the issue still persists, let's try the below methods and check if it helps to resolve the issue.

Method 1: I would suggest you run the Troubleshooter for Windows App and see if it helps to resolve the issue.

Windows 10 Apps troubleshooter: http://windows.microsoft.com/en-us/windows-10/run-the-troubleshooter-for-windows-apps

Method 2: If the issue persists, then I suggest you remove and re-configure the account in the Mail app and check.

Configuring the account should force it to sync the mails.

For reference: Mail app for Windows: https://support.office.com/en-in/article/Mail-and-Calendar-apps-for-Windows-10-Help-cc72af58-c630-43a5-8d75-bcae851565f3