I had a sort of similar issue today, I tried downloading a Sprite from inside the Project (via the image editor) but clicking the download button did nothing.
What I realised is pretty simple, it attempts to open a popup, and if you have popups blocked this will simply not work. If you use the desktop app and you get that "a popup was blocked blabla" thing when trying to connect to google drive, your popups are blocked. Interestingly you won´t get any error if you try to save a sprite or download an apk. If you go to editor.construct.net it will show a small icon on the right of the URL bar when a popup was blocked, there you can set it to "always allow". Once done the download should work just as well in the desktop app.
Hope this helps, cheers!