How To Download And Manually Install Android 4.2.2 For Nexus 10
Google has started sending out Android 4.2.2 OTA to the Nexus 10, Nexus 7 and the GSM Galaxy Nexus. If you don't get prompted to download and install the update by tapping System Update in the Settings app, you can download and manually install the latest version of Android.
You'll need to make sure your Nexus 10 is on stock JOP40D for this to work. You can check the build by going to the Settings app, then tap About phone, scroll down to the Build number and you'll see if your Nexus 10 is on JOP40D. If it is, you can now proceed to download Android 4.2.2 for your tablet.
Droid-Life has put together some excellent instructions to help you find your way to Android 4.2.2. You'll first need to download the zip file from here. Once you have that downloaded you can proceed to the next step.
Instructions (Toolkit/Custom Recovery):
1. Download the above file to the root of your device's storage.
2. To make this process extremely easy, download the Nexus 10 Toolkit.
Once the Toolkit is installed on your PC, make sure the N10 drivers are installed. You can download and install them all via the Toolkit.
3. Turn off your device.
4. Hold down both the volume buttons and then the power button. This will reboot you into fastboot mode.
5. In the toolkit, select the option to reboot into a custom recovery, but without permanently flashing it.
6. Choose either ClockworkMod Recovery or TeamWin: both work just fine.
7. Your device will reboot into custom recovery and from here, install the OTA zip file.
8. Once done, reboot and you are finished.
Congratulations — you now have the latest version of Android on your Nexus 10.