Moving my music library from icloud to Music app

Presumably you no longer have the previous computer or a backup of the same. Presumably you can copy the files from the iCloud > Music location that you can see into the local ~/Music/Music folder. Without knowing exactly what you're seeing it is difficult to give precise instructions. See Managing your Mac media libraries - Apple Community for some background on the default layout for the library. If your backup include the Music Library.musiclibrary you can hold down option as you launch Music to choose that library. Otherwise you may have to import the songs into an empty library.

See Recover your iTunes library from your iPod or iOS device - Apple Community if you cannot reclaim the songs for some reason.

tt2