How to fix the campaign progress bug (missions not unlocking)?

There a widely known bug in Age of Empires II: HD Edition which causes subsequent campaign missions not to unlock. The issue can be bypassed by starting a new game and using cheat codes to unlock levels, but cheats disable at least achievements and the method is highly inconvenient to use as missions have to be treated in chronological order. I encountered this glitch early on and expect that the incident isn't necessarily a one-off occurrence.

Is there an alternative solution?

At the time of writing the bug hasn't been fixed (all patch notes).

Just delete your current profile and make a new one, that's the only fix I know. You can use the "i r winner" cheat to quickly go back where you were before. Don't worry, if you completed every single level without cheat, you will get the achievements (worked for me). Just i r winner those missions/campaigns that you already done before.