OTA stands for Over The Air. Those update files are usually delivered wirelessly directly to your device, but as it usually happens gradually, you might get your update couple of days, weeks or even months after the rollout starts. Luckily there are ways to accelerate that process. Among all the people who get OTAs first there usually is couple of people willing to manually extract them from their device, and share with others on websites like Android Police.
You need to figure out 4 things:
After you download your OTA you’ll have a file like that:
$ <irrelevant-characters>.signed-<DeviceName>-<yourBuildNumber>-from-<newBuildNumber>.zip
Example, for Nexus 6 (aka shamu) running on Android 5.1.1 (MRA58K), an OTA upgrading it to Android 6 (LMY48T) would be:
$ 6035ff8ac2acfe9017a7dc2e7ae4bbc926bddfb4.signed-shamu-MRA58K-from-LMY48T.zip
Plug your phone into your computer and
If everything checks out, then:
$ adb reboot recovery
When you see a cracked open Android with a red exclamation point above it and it says “No command.” below, then your phone is completely bricked.
It isn’t actually. Just press Power and Volume Up simultaneously and then use Volume buttons to select “apply update from ADB”.
cd to your Downloads directory (or wherever .zip file is) and run adb sideload <file name>, example:
# replace "Downloads" and file name accordingly
$ cd ~/Downloads
$ adb sideload 6035ff8ac2acfe9017a7dc2e7ae4bbc926bddfb4.signed-shamu-MRA58K-from-LMY48T.zip
After approximately one infinity, updating and app optimizing should conclude, and you’ll be able to enjoy your new, updated system.