It seems that if you install the Launcher and go to offline mode before installing Unreal Engine, the Launcher gets permanently confused and will not allow you to download and install Unreal Engine even once you are signed in.
This procedure appears to have worked around the problem for me:
- uninstall the Launcher
- manually remove the registry entries in HKEY_CURRENT_USER\Software\Epic Games
- manually remove the EpicGamesLauncher and UnrealEngineLauncher folders in the user’s local application data folder
- empty out the user’s TEMP folder
- reinstall the Launcher and try again