The best way to ensure that album art from downloaded local files is displayed while listening on all of your devices is to make sure that the album art is 300x300 pixels. I have found that many people had luck by adding the album art to the song on iTunes and then downloading the song as a local file to Spotify, as downloads in MP3 format work best for displaying album art. You may have to wait a few minutes for the artwork to sync to the song after downloading it.
Another possible issue could be that your downloaded files have apostrophes in them, which can sometimes cause issues when adding local files. If this is the case, simply delete the apostophes from the file names and then try downloading the songs again.
It seems that adding art in iTunes consistently works well for resolving this issue, but if that still does not work, I would recommend reinstalling the songs you downloaded from your local files.