How to manually upgrade iTunes library to Music.app library on macOS Catalina?
Quit Music.app by Cmd+Q
Hold Option key and start Music.app
Choose music library dialog opens. Click Choose library...
Locate the old .itl file (e.g. Music/iTunes/iTunes Library.itl) and select it
Input a name for a new library and click Save
Music.app starts with original library contents within a few seconds. Only a few megabyte large library is created on the disk. Media files and iTunes files are left intact.
Optional: go to Music.app preferences - Files tab and check the settings, so you know which location is currently used for media files.


Note: This didn't work in beta versions of macOS Catalina – first versions of Music.app didn't support multiple libraries or selecting a library.

Good to know to avoid confusion while working with multiple libraries: double-clicking on .itl file, or .musiclibrary library (folder) from Finder opens Music.app with the last used library. It does NOT switch the library.