A typical setup is to have an AV receiver with a number of HDMI inputs and one HDMI output going to the TV.
Video and sound is sent from each device to the AV receiver and the video only is passed to the TV (video pass through).
When watching the TV with an integrated tuner, the sound can be sent back down the same HDMI to the AV receiver (Audio Return Channel or ARC).
Your problem is that even with the HDMI switcher to connect your different devices, you can only connect the HDMI output to the TV or the home theatre, not both.
Unless the TV can pass the audio to the home theatre, or the home theatre can pass video to the TV, you won't be using HDMI for both video to the TV and audio to the home theatre.