In that case I'll just describe what I did below, with the caveat that none of it may have mattered because the unlock feature spontaneously started working on its own a day or two after I had given up.
The first paragraph describes what you already did but I'll include it for context.
System Preferences > Apple ID > Overview > "Sign Out". Then, sign in again. It's the iCloud equivalent of "turn it off and on again". Over the course of troubleshooting this in the past I found it did not resolve the problem for me, but it might for you.
Then, open the Messages app and "Sign In" using that Apple ID. On that subject "Sign In" to FaceTime also, just for good measure. I know those apps are unrelated to each other or anything else but for whatever reason those actions have worked for me more than once in the past. Despite it being unrelated in any logical sense, the correlation was repeatable enough for me to surmise it may have been a factor.
Restart all devices: iPhone, Mac, Watch, perhaps your wireless router as well. Couldn't hurt.
What will not help: un-pairing the Watch and the iPhone. That will waste a lot of time and accomplish nothing.