Where does this no internet notification come from?

Try the following:

1. Temporarily disable any login startup items by holding down the Shift key BEFORE clicking on the Login button. If there is an app that requires Internet access it may be disabled using this method. Typically, VPN apps may attempt to confirm an Internet connection before it can create a tunnel. Maybe one of the apps your wife downloaded and installed was a VPN client.
2. Download, install, and run an EtreCheck report. This report may identify any issues that may be causing this. Note: The EtreCheck report does NOT contain any personal information.

• You can download EtreCheck from here.
• Start EtreCheck from a normal user account. Optionally, you can run it from a user account with Administrator privileges.
• Select a problem from the drop-down menu to enable the "Start EtreCheck" button. Optionally, you can add comments on what issues your Mac is experiencing, especially to aide others with similar Mac issues.
• Click on Start EtreCheck
• Allow the program to run to completion.
• When done, select Report from the left-side window to display it.
• Select the "Share Report" icon.

  Select Copy Report

• Paste the report to your reply post.

downloaded some apps

Download what apps? This would be a clue as to where to start.