So for me, such config is stored in this folder:
C:\Users\Admin\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn

And then there's a subfolder for each character. Stuff like:
C:\Users\Admin\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\FFXIV_CHR004000174B3A670F

And in there are the files that contain the hotbar layouts and such. So you can have a look and see if those stayed behind after you uninstalled.

Such data is not currently stored on the server but it will be in future. Not that this helps for the case in point.

There's also a tool in the PC launcher that lets you export and import the config manually, but again that's little help if you didn't back it up beforehand.