Find Apple Watch Series 8 serial number after wipe

The serial number should be engraved in one of the watchband slots on your watch. Remove the watch band to see it.

M1 Mac mini, macOS 26.3;  Watch, Series 10 GPS-only, watchOS 26.3; iPhone 16, iOS 26.3; TV 4th Gen, tvOS 26.3 HomePods, audioOS 26.3

Check your Apple ID device list, which you can do on another device or by logging in to Apple's AppleID site. If it hasn't been removed from that list, it may still show up and you can see its serial number by clicking on it.

My watch serial number also shows up on the original purchase receipt.