Hello Louis, I’m sorry for the inconvenience that you have faced with your NUC.
I would like to know if you have tried another keyboard on your system. Also make sure it is not a wireless or Bluetooth.
Which the BIOS version installed on your NUC is? BIOS version 0026 fixed an issue related to this, but I would recommend you to check the current BIOS version since you may need to install a previous one before flashing the 0026 and the latest version available 0027.
Please check the release notes on BIOS version 0026 http://downloadmirror.intel.com/23926/eng/WY_0026_ReleaseNotes.pdf
The only way to test if the issue is being cause by the keyboard will be by testing another one.
There is a BETA Bios version that you can try. This Beta BIOS is to check a potential fix for problems entering BIOS Setup with certain monitors and TVs. It enables 1776x100 when 1080i is selected and add 1184x666 when 720p is selected for HDTV via HDMI.
Please report back whether or not this BIOS allows you to access BIOS Setup with F2.