Android 5.1 is making its way to more and more Nexus devices by way of factory images, with the Nexus 4, Nexus 7 (2013) and Nexus 7 (2013) LTE now supported. Now, it doesn’t matter what you’ve done to your device’s firmware, you can restore back to a stock Lollipop image and get the best of what Google intended.
The factory images are all version LMY470 and are available today. Be mindful that, by default, these factory images will wipe your data on install and if you’re coming from anything other than a very-bloody-close-to-stock ROM, that’s probably a good thing. However, if you’re confident you don’t need to wipe, you can omit the wipe command from the update files easily enough.
You can grab the updates today from the Nexus factory images website.