For three years, I have continually asked how we can do this and thanks to the latest update from Apple, we can now use iPads wirelessly with projectors. Two directors of our IT department and senior leadership have confirmed this to be working well.
Instructions (with thanks to Professor Simon Cox who has tried this out successfully within his lectures)
Both the iPad and the AppleTV need to be on the latest version of the OS (8 for the iPad and it needs to be the most recent hardware of the AppleTV A1469 to work)
See e.g.
http://www.cultofmac.com/282209/airplay-longer-requires-wi-fi-network-ios-8/
https://support.apple.com/en-gb/HT204289
This link has further help:
https://support.apple.com/en-us/HT204291
Toggling the Bluetooth on can also help with the peer to peer iPad to AppleTV as per the link above.
Problems we have heard of have been older generation iPads. Probably works on an iPhone too if you have the latest one.
Double Digit Education 