How do I save downloaded music to my LG v20 music library?

Restart the phone (or run media.Re.Scan) to rebuild the media database. Then open the music app and look for the music. (Downloading doesn't trigger a rebuild of the database.)
 
Which app did you use to download the music? Some apps save them in such a way that only that specific app has access to it. That way they can help prevent copyright violations. If you downloaded a regular audio file like an mp3, that should show up in the music app automatically.
 

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