May be you can try to switch off the iPhone's BT, or even the phone, and then check the Airport again.
Yep tried that. As soon as I disable BT on the phone the watch disconnects (I get the red disconnected symbol) and nothing works anymore, not even things that are supposed to work like Siri, phone calls, etc. As I said before, the watch would not show up as WiFi client on the Airport, nor could it be seen by IP Scanner Pro and the like.
The good news is, it's not the watch that's broken #
I set up a new AP using a spare Airport and the watch uses that WiFi network just fine. I can use Siri and make phone calls with BT on the iPhone turned off. Phew!
I believe I have tracked this down to my very long WPA2 passphrase. This has been known to cause problems in the past, for example I had trim it from 64 to 63 characters for certain Windows clients. I cut it in half, now the watch connects.
So with this all fixed I can confirm that the watch does indeed connect to a WiFi network known to the iPhone. I apologise for making confusing statements based on experience with just my own watch and my own WiFi network at home, which at the time would;t play nice together...
To the OP, I haven't found a way of displaying which network the watch is connected to (other than using a sniffer or looking at the Airport client list).