I've had the same problem over the last few days. Had my Moto 360 for 3 weeks now with not a problem at all, received the Android Wear update and I had similar results as reported online - better battery life, more stable Bluetooth connection, etc.
Then the time on my watch started going awry, moving forward in I think 4 minute intervals. Only restarting the watch helped, but of course I had to double-check the time on my phone every time I looked at my watch to notice. Sort of defeats the purpose of a watch, smart or not!
And NOW just this evening my watch had stopped responding to core smartwatch functionality; I still receive notifications / emails from on my watch from my phone, the time is currently correct on the watch, and I can perform random Google search queries ("Ok, Google ... who is the Prime Minister of Canada?") and it reports back just fine.
But - I can no longer set a reminder, for instance. "Ok, Google ... Remind me to <blah blah>" and it says it can't reach Google. But it can! Because I can perform Google searches from the watch, and I can say "Ok, Google ... Set an alarm." and wait for the prompt. But if I say "Ok, Google ... Set an alarm for 9:30pm tonight" it just pops up a Google search that I can open on my phone.
Again, I've been using my Moto 360 CONSTANTLY for the last 3 weeks with none of these problems. And now I've gone through the reset cycle several times since the time on my watch started being incorrect a few days ago ... I clear data on Android Wear on my phone, uninstall AW, turn off Bluetooth. Reset device on my watch, then follow on-watch prompts to reinstall AW on my phone, re-pair Bluetooth, and so on. Ad nauseum. The watch just isn't working properly anymore regardless of going through complete reset cycles of my phone and watch both. AND I've contacted Motorola tech support a few times now, they admit I've done everything correctly and there's nothing more to do than exchange the watch. Which I will do tomorrow, but don't expect any changes as this is obviously a software problem that should have gone away with a full device reset. Argh. I really liked my Moto 360