I got it working ok, including with the Radeon graphics card. Took some doing.
Setting up a NUC8i7HVK with Ubuntu 18.04 with AMD VEGAM graphics
I love my NUCs, and when I gave my mine to my daughters to watch
  movies and play games on, I had a perfect excuse to purchase a shiny
  new Hades Canyon NUC (NUC8i7HVK) with an AMD Vega M graphics card.
  Little did I know GNU/Linux wasn't supported on it; I wrongly assumed
  that all NUCs were compatible with my preferred OS. I'm not gonna use
  Windows, so I was forced to put some time into setting up my NUC.
Thank goodness for the really smart people who actually figured out
  how to do it; I'm just documenting the steps I took trom their advice
  so that others can do the same (I've also linked to pretty much every
  page where I found good advice on this problem). The keystone advice
  came from user834610 on
  this
  page and from a bunch of people
  here.
Note: My Hades Canyon NUC is the one with the i78809G CPU (the more
  powerful of the two available options). From what I've read, at least
  one of the steps below may fail on the HC NUC with the other CPU. Fair
  warning!*
Update firmware
- Download the BIOS file from here
  (use the one for the F7 BIOS Update method)
- Update the BIOS using these
  instructions.
Install Ubuntu 18.04 from a USB stick
Make an Ubuntu startup media on a USB flash drive. I was about to link
  to the instructions for doing that, but if you don't know how to do
  this already, it is probably not a good idea for you to continue down
  this road; it gets a little hairy if you're a newbie to GNU/Linux!
  This NUC is not the place for your first Ubuntu rodeo.
- Plug in the USB startup media and fire up the NUC. It won't work. It'll show you the traditional choices (try Ubuntu, install Ubuntu),
  but no matter what you choose, you'll get a black screen.
  
- That's because the Linux grahics card drivers on the live media can't deal with the hardware. You need to dumb everything down by
  telling the kernel "nomodeset," meaning it's not allowed to start
  video drivers until the system is running.  
 
- After turning on the NUC, the moment you see the Grub screen (the try vs install options), press 'e'. That'll get you to a screen where
  you can configure the boot options.
  
- Replace the words 'quiet splash' with 'nomodeset'. A bit like this
  but actually removing the 'quiet splash' (because instead of a pretty
  splash screen now you'll see what's actually going on - that's the not
  'quiet' part).  
- Press Control-X to exit and boot. Now it should work.  
 
- Go through the usual process of installing Ubuntu.  
- When it finishes, it'll fail to boot again, since the newly installed Ubuntu doesn't have the nomodeset parameter and will try to
  activate the ungovernable video hardware.   - 
- Do the whole nomodeset dance again. Here
  is a pretty good explanation of how to make the nomodeset option
  persistent (edit the /etc/default/grub file to add the nomodeset and
  then run sudo update-grub2).  
- I actually just booted, hit Control-Alt-F3 to get to a tty terminal instead of going to the GUI environment, edited the
  /etc/default/grub file (changed the line
  GRUB_CMDLINE_LINUX_DEFAULT="quiet splash"toGRUB_CMDLINE_LINUX_DEFAULT="nomodeset", then ransudo update-grub2and rebooted. That worked and maybe saved a minute or two.
 
- You should end up with a working installation, but you'll notice that you can't change the display parameters, the HDMI sound output
  may not work, and if you try GLmark2, GLXGears -info, or glxinfo,
  you'll see that there's no hardware accleration. In other words,
  you've just put all of your hopes and dreams into the NUC's graphics
  card in vain. You are where the person who asked
  this
  question was!   
Now comes the tricky part. In order to get the graphics drivers
  working, we need to:
- Upgrade the Linux kernel to 4.18 or higher  
- Grab the vegam firmware blobs needed to talk to the hardware  
- Update Mesa to at least 18.1  
Update the kernel
Ubuntu comes with a frozen kernel. Version 18.04 Bionic Beaver comes
  with Linux kernel 4.15, and that's what you get. The drivers for the
  AMD GPU come with Linux 4.17, and from what I understand serious
  support comes only with 4.18. In any case, you'll have to upgrade.
You can do that manually like this:
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v4.18-rc5/linux-headers-4.18.0-041800rc5_4.18.0-041800rc5.201807152130_all.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v4.18-rc5/linux-headers-4.18.0-041800rc5-generic_4.18.0-041800rc5.201807152130_amd64.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v4.18-rc5/linux-image-unsigned-4.18.0-041800rc5-generic_4.18.0-041800rc5.201807152130_amd64.deb
wget http://kernel.ubuntu.com/~kernel-ppa/mainline/v4.18-rc5/linux-modules-4.18.0-041800rc5-generic_4.18.0-041800rc5.201807152130_amd64.deb
sudo dpkg -i linux-*.deb
But I cheated and just used UKUU.
sudo add-apt-repository ppa:teejee2008/ppa
sudo apt update
sudo apt install ukuu
Ran UKUU from the GUI, chose the Linux Kernel 4.18.3, rebooted.
Of course, it failed to boot.
Because I had to go into the BIOS setup of the NUC and disable Secure
  Boot.
- On startup, press F2 to enter the settings, and set
  
- Advanced -> Boot -> Secure Boot -> Secure Boot Config -> Secure Boot = unchecked
- like this
 
After disabling Secure Boot, Ubuntu came up just fine, and running
  uname -a showed that I was now running the 4.18 kernel.
Upgrade Mesa
sudo add-apt-repository ppa:ubuntu-x-swat/updates
sudo apt dist-upgrade
Grab the AMD Vega M Linux driver and put it in the appropriate directory
wget -m -np https://people.freedesktop.org/~agd5f/radeon_ucode/vegam/
sudo cp people.freedesktop.org/~agd5f/radeon_ucode/vegam/*.bin /lib/firmware/amdgpu
Then update your Initial Ramdisk to recognize/choose the right kernel:
sudo /usr/sbin/update-initramfs -u -k all
Turn off the nomodeset option again
- Change the relevant line in /etc/default/grubtoGRUB_CMDLINE_LINUX_DEFAULT=""
- Run sudo update-grub2and reboot