On iPad Mini, AppleTV doesn't work with Google Hangout video chat
I googled around for it, but all I found was this question asked with the same words on other fora.

The second link has someone giving the answer I thought likely: the app disallows AirPlay. Some app makers don't want you to watch their iOS content on your TV. HBO Go was a notable example, but they updated their app to allow AirPlay streaming to AppleTV in February 2013 (see here). While I can't find any documentation that the Google Hangout iOS app does not allow AirPlay, the behavior you describe is at least consistent with that hypothesis.