My first time on Communities but I had to reply because this issue really bugged me until I read the suggestion from MattBoxer.
I had this exact problem with the Apple TV app not having any audio when played through my Chromecast with sound through a bluetooth Jambox.
I went into Chromecast settings (click on profile) > Display and sound > Advanced sound settings > Select formats
and then select "None: Never use surround sound" and that fixed the problem.
That is, the sound started coming through the Jambox. I don't have a surround sound system, so if you wanted to use both (say if you wanted to use both surround sound and bluetooth headphones at different times) then that may be a problem? Maybe you'd have to switch surround sound back on when you wanted to use it.
Interestingly, the audio problem occurred when I tried to play my purchased series of the Sopranos, but not for other movies and things I had purchased on iTunes. For example, the Ken Burns Jazz series audio worked before I turned surround sound off. So maybe different programs have different audio settings?