Google has still not released the factory images for Android 5.1 for the Nexus 6. However thanks to the development community you can run Android 5.1 on your Nexus 6 now. Developer "Stephen" has released his own build of 5.1 LMY47D for the Nexus 6. This should be considered a test build for now. Google has still not released the Vendor blobs for 5.1.0 so there are a few bugs in this build. Issues include no working camera, and phone calls. Phone calls will work but apparently for now to hear the other person you will need to press the "speaker" button.
Features include Encryption disabled by default, disabled recovery so that the stock build won't automatically overwrite your custom recovery, Android L wallpaper, updated Android L phone icon, android L power menu, android L bootanimation, and some other minor tweaks. The developer suggest to mount the system when installing. Grab the download from the link below.
via XDA