I searched for hours and hours before finding this StackOverflow answer, which got me on the right path. If your app has a Jump List and it's appearing fine when right-clicking on the app's icon in the task bar, but it doesn't appear in a pop-out menu on the Start Menu, it's possible that this is because your start menu entry is a shortcut and you need to set that shortcut's AppUserModelID. If you don't do this, Windows doesn't realise that the shortcut has anything to do with your app.
If you're using Inno Setup as an installer, you just need to add an AppUserModelID: tag to the entry under [Icons] that creates your Start Menu icon.
Hopefully posting here will make the answer more easily Google-able for others.