Lightroom catalogue disappeared after uninstalling light Lightroom Cloud based app

I don't know where LRc backed up to

Use your operating system's search feature to search for all files whose name ends with .zip. Some of these found will not be LrC catalogs, but some might be. Often the default name Lightroom 13.zip (or something like that) will make it obvious.

Regarding the fact that you don't know where your backups are stored. ADVICE: you should never be in the position where you don't know where the backups are. From now on, you should know, no excuses. ADVICE: store them in locations that are easy to remember, and make use of the fact that you can make folder names descriptive and meaningful. My Lightroom Backups are on my external drive, in a folder at the root level, named Lightroom Backups. Now there is never any doubt in my mind where my backups are, they are easy to find, I don't have to remember a default location, I don't have to remember some folder location that is four folders deep. By being intentional and obvious about the use of folder names and the location of the folder, you won't have this problem in the future. Use folder names and locations to your advantage!