Check the status of a Finder copy-paste operation from Apple Watch
I doubt such an app exists even for iOS, leave aside watchOS, for watching Finder file operation progress. This is a very specific use case and I am not certain if there are any official APIs that lets you obtain this data in a watchOS or iOS app.

A simple solution that I can think of is to install a VNC viewer app on your iPhone (many available on the App Store). Configure your Mac for screen sharing. Now using the viewer component on your iPhone/iPad, you can visually see the screen of your Mac and check file operation progress.

VNC apps are not just viewer and you can also control your Mac from your iPhone, thereby giving you the ability to perform operations on your Mac remotely.