Use FindMy on Apple Watch to find iPhone

It already does: Set separation alerts in case you leave a device behind in Find My on iPhone - Apple Support. Then just use Find My in the Apple Watch to locate your iPhone.

The app on the Apple Watch is actually called "Find Devices", not "Find My". Don't ask me why :)

I see that you participated in this discussion: "Notify When Left Behind" not working on … - Apple Community. The conclusion I draw from it (which might be incorrect) is that the Apple Watch needs a cellular connection to be able to receive a notification. But, somehow, it doesn't make sense; Apple makes no mention of this requirement in the article I cited above.

This recent discussion could perhaps interest you: "Notify When Left Behind" not working on … - Apple Community.

I doubt it. I'm not sure what's going on. You could try to unpair and pair again your watch; this can help with pesky issues between iPhone and Apple Watch.

When pairing again, you'll have a choice to restore a backup of the watch. Do this first. If the issue persists, unpair and pair again but choose "as a new watch." You'll loose all your previous settings on the watch but it will be like your watch is brand new to your iPhone, which may clear the issue.

I admit I never had the chance to test the functionality on my Apple Watch when I leave my iPhone behind but I can tell you that in my experience with AirTags, when one is left behind, the notification only pops up 5 minutes later. Perhaps it is the same about of time when an iPhone is left behind.