I have got a brand new Nexus 6, and brand new LG G Watch. Everything seems to work fine, but when I try and pair them, at the final stage the watch says "Just a minute" for a long time, and then finally goes to the dialog saying "Can't connect to the Internet. Swipe for some tips."
I've followed all the tips at g.co/AWhelp (https://support.google.com/androidwear/answer/6058728?rd=1) but nothing seems to work.
The only thing non-standard about my set-up is that the phone has no SIM card. I wanted to play with Android Wear development, and so it is a test/development device.