I contacted gog's support. Here are a few things they told me to try out (none of them solved the problem, but I'm copy-pasting them for future reference):
Please try these troubleshooting steps:
1) Please make sure that you use "Run as Administrator" to install and to start the game (right-click on the shortcut or setup file and select "Run as Administrator"). If you are using Galaxy, then make sure to run Galaxy as Administrator.
2) Please reinstall the game AND add it to your anti-virus/firewall software's exception/trusted list. Simply disabling firewall or AV software could not be enough, since some of these software are running still in background or in services list. If you are using Galaxy, then please add Galaxy as well.
3) Reinstall the June 2010 DirectX runtime package. You can find the archive file containing the 2010 package installation files here: http://www.microsoft.com/en-us/download/details.aspx?id=8109
After that please download and install the DirectX Web Installer: https://www.microsoft.com/en-gb/download/details.aspx?id=35&e6b34bbe-475b-1abd-2c51-b5034bcdd6d2
You will have to extract the files into some folder, then open that folder and launch the DXSETUP.exe file in order to proceed with the installation.
4) Maybe you have some settings forced on your graphics device in NVidia Control Panel. If so then try to reset them:
I) Right-click on your desktop and select Nvidia Control Panel.
II) Click Manage 3D Settings.
III) In the upper-right corner, click Restore Defaults.
5) Try disabling DSR: http://www.gog.com/support/support_view/potential_issues_with_supersampling_dsr_factors_feature_on_highend_geforce_cards
and from a follow-up mail:
"Please make sure that no third-party applications are interfering with the game. Please try to install and run the game in “clean boot mode”. Instructions on how to do that can be found here: http://support.microsoft.com/kb/929135
Make sure the account you create for clean boot mode is granted Administrator privileges."