Your Nexus 6 will not install OTA (over-the-air) update if the phone is rooted or the bootloader is unlocked. Even if you have uninstalled the root and custom recovery from your phone, the OTA updates will not install on it. This is because of changes to the system image made by root and custom recovery. There is a workaround to resolve it.
Checkout the below guide to know how to install OTA update on a rooted Nexus 6. Thanks to the XDA user bitpushr, he has discovered a method to install OTA (over-the-air) updates, even if your Nexus 6 is rooted. You only need factory image and corresponding OTA system image in hand.
This guide is going to assume that you are already pretty savvy with tweaking Android OS and that you have got Fastboot and ADB installed on your computer. Fastboot and ADB is not installed, follow this guide.
Note: Before downloading, go to ‘About Phone’ and note the first 6 letters. It should say something like LVY48E. Download the same system image with the same letters.
2. Once downloaded, extract the system.img file using 7Zip and transfer all contents to the ADB installation folder on your PC.
3. Now, download the right OTA update file [HERE] for your Nexus 6. The file should be an upgraded number from your current version. Otherwise, it will not work.
4. Now, turn off your Nexus 6 and boot into bootloader mode by issuing the below command (Open a command window inside the ADB folder and issue the command)
adb reboot bootloader
5. Now, flash the stock firmware by issuing the below command:
fastboot erase system
fastboot flash system system.img
Your Nexus 6 is now back to stock and unrooted. Next step is to flash the OTA file and root your phone.
1. Turn off the phone and boot into recovery. Select ‘Apply update from ADB’.
2. Open command prompt and flash the OTA file by issuing the below command (change OTA.zip to the name of the file you are flashing)
adb sideload OTA.zip
3. Once it is done, your phone will reboot. You have successfully installed an OTA update on a rooted Nexus 6.
If you want, you can now root your Nexus 6 again by flashing SuperSU update zip file. (Source – XDA)