apple tv 3rd generation and iPhone iOS 12.0, airplay, screen mirroring

That would be more of a network issue

Make sure no other activity is occurring (other streaming, web use, network backups etc)

If on wifi try ethernet

To see a current status of the network get a report from stumbler, netstumbler or similar. Look for signal strength, noise, nearby networks

Test on another network

Just make sure that both devices are compatible with AirPlay and have been updated to the latest version. Check solutions here to fix it.

https://airmore.com/fix-ios-12-screen-mirroring-not-working.html#_2

Sorry, shouldn’t have been rhetorical as it very well may be a shared network issue.

I tried to connect in the same network and with the Apple TV hooked to WiFi and it still did not work. It is in a school setting and have other network technicians looking into it but the problem has shut down how I present information to my students.

I have the same issue, I tired from both Ipad[Version 12.1] and Iphone. Airplay detects the Apple TV and awakes it, if the apple tv is in sleep. Then I stop the mirroring, tried to use apple tv app on both devices. I keep getting "Looking for Apple TV". Something broke after the update.

It will be up to the network techs to fix it as that is where the issue lies.

Ok but everything was working great yesterday when I was using iOS 11. How is this on them?

Issues can come at any time. Airplay is done over the local network and when there are issues that is where one needs to.look

It wouldn't be due to an IOS update, the 3rd Gen is discontinued and hasn't had an update since 2016

You would need to troubleshoot as above