I found an Apple Watch on a bus. The battery was dead so I charged it. There have been no contact efforts from the owner. What should I do?

Thank you for wanting to do the right thing and reunite the watch with its rightful owner. So, good on you.

As mentioned by others, Apple doesn't get involved in lost or stolen devices. And since it's more likely you don't know the Passcode for the watch, you don't have much chance of getting into contacts on the watch as recommended above.

So, turn it over to the local authorities in your town and hope whoever lost the watch filed a police report. It's totally useless to you anyway.

hand it in at a local lost and found or your local law enforcements, apple don't get involved

If you can get in to the watch, you could go to Contacts and see what is there.

Neal