Nexus 6 failed to update OTA 6.0.1 from MMB29V to MMB20X
First off I'm assuming that the device is already flashing unlocked since you said you tried to flash the update from recovery. You shouldn't need a backup but make one anyways because recovery might be flashed over. Read everything first before you get started.
Download current image (the one you have now, you can try 29X or latest 30D), it's best to restore your system then update for guaranteed results and no misbehavior after reboot (I'm a little adventurous so I'd go for the 30D). unpack the .tgz and cd into the resulting directory.

fastboot flash radio radio.img
fastboot flash bootloader bootloader.img

Note the images aren't radio.img and bootloader.img, just do radio and tab and bootloader and tab.
Unpack the .zip that was extracted from the .tgz. cd into the resulting directory.

fastboot flash boot boot.img
fastboot flash system system.img
fastboot reboot

This should maintain all your data on the device after you reboot and I'm about 99 percent sure it'll work. So you shouldn't be needing to backup. If you run into problems after reboot you'll need to flash cache as well.
Reboot into bootloader again and use

fastboot flash cache cache.img
fastboot reboot

If it still doesn't work, you'll need to have a backup, reboot to fastboot and use the flash-all.sh that was extracted from the .tgz file. This would delete all your data but you can flash recovery again with fastboot flash recovery &lt;image filename&gt; and restore Data from the backup you did.
PS. If you have to flash cache you might lose some data but only relating to system apps and services.