Let’s try to narrow down the issue. It is necessary to use high quality straight video cables, no adapters. Then, we can test the system using only the NUC, 1 monitor and 1 memory stick of 1333 or 1600MHz at 1.35 Volts. Standard notebook memories of 1.5v do not work.
Turn the NUC on, check if the color of the power button light changes of color. Usually is red and after turn it on the light changes to Blue. Verify if the video input of the monitor is properly selected (HDMI or DisplayPort).
Let me know if you can see the Intel logo splash and the options F2, F7 and F10 or any other information in your Display.