Help with restoring catalog location for PSE 18/19

Why is it more urgent in your case to search for 'catalog.pseXXdb' files rather than .tly ones? (as suggested by 99jon and Greg)

The backup.tly file only exists in a backup folder, never in a normal catalog. From your description, the only one in your computer is in the backup folder.

By default the catalog is a folder located in a hidden location in the C: drive. There is a small chance that the latest catalog folder has not been wiped in the reinstallation of Windows.

Without such a catalog folder, the best you could do would be to restore the catalog from your last backup, on 'original location'. That should not delete all the files imported after the last backup, but you would have to re-import your library, knowing that already present files in the catalog (most of them) will be skipped and not imported as duplicates. If you have 'written metadata to files', you'll recover your media tags. Do a test with reimporting a relatively small folder tree to get acquainted with the process.

. . .  or search your computer for a file named catalog.pseXXdb, where XX=16 or 17 (the version numbers of Elements 2018 and 2019, respectively).  Make sure that the File Explorer View menu has the File name extensions box checked.  Then double-click on the file.  The Organizer should then open with the catalog. 

The calalog data base is usually found in this location:

C:\ProgramData\Adobe\Elements Organizer\Catalogs\My Catalog

Would you possibly have a back up of the catalogs folder. If you can find the named catalog folder the catalog database should be inside – see image (you may have version 16 or 17). It can be opened with PSE Organizer.exe