This is happening to me too. Here's everything I've tried:
1) Running it compatibility mode. None of the Windows versions worked, and weirdly enough Win98 compatibility wouldn't even launch because it was looking for a CD.
2) Enabling all the compatibility options. No matter what combination I tried it didn't seem to help. At best, I could make the mission loading screen last a little longer before it faded to black.
3) turning off Shadowplay. I know sometimes recording software like Shadowplay can mess with old games. No dice.
4) In-game graphics option. Even after turning everything either off or to it's lowest it still didn't work.
5) installing the game to C:. No help.
The menus run fine, and I can do everything there, but as soon as I start a mission it fades to black, then slowly the graphics load in, but the game is unplayable. Trying to move the mouse or camera is impossible. I really hope this gets fixed, since I'm a big fan of WH40k, and I've always wanted to play Chaos Gate.