Thanks Jason for your answer.
However, I have tried your workaround when I was running (K)Ubuntu but it didn't work for me.
It seems that, except for you, only OpenElec works well for HDMI sound according to what I have seen here and on other web sites. But, OpenElec is too limited for what we can do with this NUC.
I had the same problem with Lubuntu 14.04 and the above solution did not work for me either. In the end I solved it by installing pavucontrol (PulseAudio Volume Control) which gave me the option to select the HDMI output.
When installed I found it in the Lubuntu menu>Sound & Video section and launched it. Under the 'Configuration' tab I chose Digital Stereo (HDMI) Output. I believe you can also launch it by right clicking the volume icon in the task bar and choosing 'Volume control settings'.
I cannot believe the HDMI option is not switched on by default or made more easily accessible!