According to a post on AndroidWorld (via Ausdroid), the reason why some of LG’s wearables are not supported is because the functionality was not built in the first place. Based on the translated statement from LG, “Regardless of what HAS BEEN Reported, the latest version or [sic] Android Wear does not automatically providence [sic] Wi-Fi support Unless the device was designed with this functionality in place at the time of sale.”
The statement claims that the LG G Watch was not designed with WiFi in mind which is why it will not gain the compatibility, but the LG G Watch R has and a maintenance update is required in order to activate it. LG claims that its engineers are working on the patch which should be released in the third quarter of 2015.
In the meantime the LG Watch Urbane appears to be supported so if you have the watch or are planning to get your hands on it, you won’t have to worry about WiFi compatibility in the upcoming Android Wear update.