As what was explained in the solution happened to me. I was prompted to enter the CD key, but had saved it to a text file on my hard-drive so cancelled and exited the game. I then wrote down the key, to then go into the game and find that I'm now no longer getting the prompt, and therefore unable to enter the key.
I don't feel secure about this because the way the Settlers 3 "Copy protection" works is by altering things in the game, that are not obvious at first glance. For example, the Iron smelters randomly stop making iron, instead they make pigs. With this happening, its not possible to build an army!
Ideally I want to know (without spending hours playing to find out) if the game is actually in "genuine" mode and not thinking its a pirated game, and all kinds of weird things happening.
If I'm not mistaken the copy protection was not dependent on the serial key, but only kicked it if a non-original CD was used for playing and/or installation (parts of the original CD were made "damaged" on purpose as part of copy protection). 2 things were altered: aside from the iron smelters making pigs which you mentioned (pig iron, get it?), the trees planted by foresters wouldn't grow properly, so the woodcutter couldn't cut them and they'd only take up space without providing any wood.
I have the CD and not the GOG version of this game, so I can't answer your question, however I'm almost 100% sure that the GOG version doesn't have this bug because:
Hundreds of users with gog versions have posted questions about the campaign on these boards. If they had the bug they wouldn't have been able to play the game. And at least one of them would have mentioned it.