If you were to put it in a dock, I would imagine that dock would also charge the controller which would be the only reason the uScreen would keep the screen on in the first place. If it wasn't docked, then the uScreen would go to sleep to conserve battery power just like the Wiimotes.
I imagine that there will be dock, and while the uScreen is docked, not only will it charge, but by default, the console will then be allowed to go into a minimal power mode (if you have set up online with WiiU Connect 24) where the console will continue to stream to the uSCreen but have most of the CPU cores shut down and the GPU clocked waaay down until they are needed.
While in docked (non-use mode), the uScreen will show a digital clock with a status bar at the top that will display if you have any messages sent to your system, and games or update downloads currently in progress, battery level, the date and a much smaller clock for when the main screen gets used for other stuff.
Below the status bar at the top, there would be your friends list on the left or right (put it where you want) that updates live as friends log on or off and it will also tell you if they are playing a game and what game they are playing (privacy setting dictate what will be shown and to who).
Just by tapping on a friend, an option screen will pop-up allowing you to chat, send a message, send a game invite (if you are currently in a game) or send a request for video chat. If you need to type a message, then an onscreen keyboard will pop up and you can type on a full touch QWERTY keyboard.
There will also be a few touch buttons/widgets at the bottom of the screen that will provide more information. Things you will be able to put as widgets at the bottom of the screen will be apps like Weather, Media Player & Picture Frame/Slide. You will also be able to put app shortcuts, such as Internet Browser, VC Games, WiiUWare(games and apps), eShop, Netflix and an icon for whatever game/movie is in the Disc Drive.
(As one of your options, you could just make it look like the Wii main menu with a bunch of channels, your clock at the bottom and your status bar at the top)
*You will be able to message, chat, video chat, internet browse, play certain WiiUWare games and watch Netflix without unDocking the Screen*
**Docked in-use mode obviously uses more power than docked non-use mode**