Q: Does the US Apple Watch go into 24 hour time mode?

There is a clock setting in the apple watch app on your iPhone that should let you switch to 24hr format.

On a somewhat related note - is there any way to display distance walked from the Actiivty app in km rather than miles if your Language and Region is set to US?

Go to the workout app on your watch and select one of the workouts like "outdoor walk". Swipe to the page where you can select distance. Force touch the screen and you will get the option of setting it to "KM". I'm not sure if the setting sticks from then on or you have set it every time.

Those do show up as km, but the Distance shown in the Activity app on the phone, right beside the steps, only shows up in miles.

The only thing this does is to let the digital time display go to 24 hour mode. There is no 24 hour purist analog clock face, nor any sort of GMT- like complication, a la Glycine Airman, etc., available. So, the 24 hour mode is sort of a joke.