Thanks for joining the NUC community.
I understand you are not getting any audio out of the NUC when connecting your external hard drive that contains the operating system.
I know the audio has been tested and works properly when installing an internal SSD drive to the proper slot. All drivers were installed and the NUC gets audio without any issues.
Have you tested your NUC with internal SSD? Does your NUC have sound?
The graphics controller on this NUC supports HDMI 1.4, try testing the NUC with 1.4 HDMI cable. Also make sure you have an updated video driver, this will install latest HDMI codec on your computer.
Please filter by operating system and try to install the driver when it finishes downloading, to verify that your system will accept it.