The Google Nexus 6 for T-Mobile is now receiving the Android 5.1.1 OTA update. The update's approval was confirmed by the operator's product evangelist Des on Twitter and Google+ a few hours ago and it has appeared on T-Mobile's product support pages.
The OTA brings the Nexus 6 from build number LMY47M to LYZ28E, it should be around 125MB in size and have T-Mobile WiFi calling on board as well as all the bug fixes and improvements that come with Android 5.1.1.
As usual, the update will be rolled out in stages, so it may be a few days before it is available for your device. Once you do receive the OTA notification, ensure your battery is appropriately (at least 50%) charged, and that you are connected to a Wi-Fi network before downloading and installing it.
You can manually download and install the official OTA file for LMY47M -> LYZ28E and follow this guide to install it on your Nexus 6. You could also consider waiting for Google to release the Android 5.1.1 factory images for your Nexus 6, if you want to perform a clean install.
source - @AskDes | +Des Smith | T-Mobile