LG G3 owners on T-Mobile should expect an update to begin rolling out to them, containing a fix for data roaming issues, a new monthly Android Security Bulletin from Google, some stability fixes for LG's built-in apps, and some tweaks to general system stability. The update bears build number D85130g, and weighs in at 132MB. On the security side, the phone is being brought up to the June 5th security bulletin level. Those who want to keep the full system software handy in TOT form in case their phone ends up needing a factory restoration with LG's flashing software can expect to download 1.1GB.
Unfortunately, this update does not contain Android 7.0 (Nougat), and such an update does not seem to be on the horizon for the aging device, at least on T-Mobile. A version of the device popped up on Geekbench not long ago, seemingly being tested for an official Nougat update, but nothing has come of it thus far. The phone is three years old at this point, so chances of an official update in any capacity are slim to none, meaning that those who aren't willing to root the device and install a custom ROM will probably have to get their hands on a newer device in order to get a taste of Nougat.
This update is relatively small, but all the usual caveats apply; having the device lose power in any way during the update could be disastrous, and could even render the device fully inoperable. In order to avoid a potential hard brick, ensure that your phone's battery is at 50% or higher when you start the update, or that it's on a charger, and do not remove the battery or turn off the phone until the update is fully finished and your device boots back into Android. The update should be rolling out to all users, so if it doesn't come to your device automatically, try checking in the "About device" section of your settings, and if that doesn't bring the update your way, rebooting the phone may do the trick. If neither of those methods get you the update, you may want to simply wait a little while longer for it to roll out to you.