I'm going to copy and paste from the FAQ at Celestial Heavens, found here:
Question: After finishing a campaign, I can't move on to later ones; there was a check on the picture but now it's gone. What's wrong?
Answer: That's a very common problem. Here's how it works: H3 can't remember what campaigns you've finished "on its own", but it can remember it in a savegame. That's why, after you've finished a campaign, you're asked to make a save. If you wish to continue with another campaign, you have to go to that savegame and load it, and you should have the campaigns you've finished marked as such. If you want completed savegames where all the campaigns are finished, give us [meaning the Celestial Heavens forum, though there might be someone on GOG that can help for all I know] a shout, and we'll see if we can help you out.