I ran into a really annoying graphics issue with the GOG version of CMBO. All of the in-game text was rendered gray with a black background. The demo that I downloaded didn't have this issue. The text should render white on a transparent background.
After doing a bunch of troubleshooting and such, I narrowed it down to the actual art assets in the bmp folder of the game. It looks like the GOG version is using different images than the demo version that I tried.
The font images are named 30000.bmp, 30001.bmp, and 30002.bmp. To fix the issue, just download the demo from here...
http://www.battlefront.com/index.php?option=com_content&task=blogcategory&id=66&Itemid=107
And grab the above files from the bmp folder and copy them into your GOG version's bmp folder.

could you please upload the files somewhere, since I can't download the demo from the site?

The files are attached to this post, but you will have to change the file extensions from .jpg to .bmp
Let me know if it works, I have not had time to play the game yet

The choice of background and font colours is really weird.

Could a staff-member fix this permanently, please?

Copying the files from the demo is one solution.

This somehow doesn't work in all game menus for me. The battle setup screens before the actual battle are still unreadable. I'm using a Radeon Mobility 9600/9700.

I don't have issues with the Mobility Radeon 9700 in the setup phase after replacing the three files.

Which screens are you talking about?

I mean all game menus except the main 3D gameplay screen. Text appeared white on white or black on black, IE completely unreadable.
I've found a workaround. I needed to replace the three .bmp-files here with the files from the demo, or the "originals" in the radeon text fix archive available on battlefront.com. It seems gog.com have applied the Radeon text fix by default, although it's only working for certain Radeon graphics cards (and apparently not mine), which causes problems for all users that don't have those specific cards. Battlefront have removed this fix from their website, I think because the problem that was supposed to be fixed has disappeared in newer ATI/AMD Catalyst versions.
So, now to the actual solution:
After replacing those three files, I still had errors on the text in all menus of the game except in the actual gameplay screen. But it can be fixed by just alt-tabbing out and into the game. BTW, I've tried the Combat Mission Barbarossa to Berlin Demo, and it doesn't have this problem. I think gog.com should do something about this.

You can customize 3 bitmap files in your circumstance.
1.download and install GIMP.
2.open font file[30000.bmp,30001.bmp,30002.bmp].
3.Select>Select by Color,and turn off all checkbox,threshold to 0.
http://docs.gimp.org/en/gimp-tool-by-color-select.html
4.left click font or background then all same color pixels are selected.
5.change color.you can make background transparent with Color>Color Alpha.
http://gimp.open-source-solution.org/manual/plug-in-colortoalpha.html

CMBO font fix
I modified all three bitmap font files at which font colour to black and background to
transparent.

Cool. Gog should modify their installer, because most users will experience this problem.

Thank you wilaboli. Downloading the two image files you attached and changing their names to 30001.bmp and 30002.bmp and putting them in the BMP folder of the CMBO game worked. Please note that when you download the files they have a long alphanumeric string, presumably added by GOG.

For some reason I still can't get these fonts to show up in my game. It's still showing all white blank with no black lettering.