I had the same problem and took my watch to the Apple store. The "geniuses" there knew absolutely nothing about the watch or it's workings, which was a surprise to me.
They finally brought out a watch that they knew was working and compared the pulse to my watch. All agreed that mine was virtually non-existent. So, they sent my watch off to repair, sort of like sending it off to "the black hole of Calcutta." Once it goes there, there's virtually no way to speak to anyone to find out how the repair is going, are they going to replace it or try to fix it, when will I get my watch back, etc.
Here's what I learned: Early failures are being replaced, albeit on Apple's timeline. Typically, that takes 3 -7 working days. I got mine back in 7 days, but only after raising quite a pronounced stink - never rude or mean, just letting them know that I wasn't going away without some useful information The Apple store manager stepped up and went to bat for me with Support, trying everything he could to get an approximate ship date. Then, finally, I got a message that my watch had been shipped to me. Now, it was a matter of waiting for it to surface at the store. It showed up at days after I turned it over to Apple and within 1 hour of arriving via FedEx..
Here's a critical piece of info: The haptic pulse will not work unless the iPhone is in locked mode. You'll still l get the message on the watch in text form, but no haptic pulse or buzz.
My hope is that this will be a fix in the next software or firmware update.
Good luck if you send your watch out for repair, and don't hesitate to be persistent in your inquiries. Squeaky wheel, etc.