I think the apple watch is perfect for biking! I ride on and off the road and use it every day. The apple activity app that comes with the watch will look for you wrist but if you use one of the other cycling apps the watch works great! Apple watch can connect via blue tooth to a heart rate monitor (and a host of other devices like powermeters etc) so it is easy to log all my rides and monitor distance speed and other data directly from the watch. I just need my phone in my jersey or backpack and the apple watch mounted on my handlebars. I can also control my tunes without pulling out my phone. You might want to check out a mount for the handlebars like this one: http://velomounts.com/
Another nice thing is that I just need one device -no need to have more than one gadget attached to the bike. I just put the wristband back on when I get off the bike! Once again I have found a way to use an apple product I could not have imagined a year ago!
PS I am able to receive all notifications like text email and phone while riding, so not sure why yours does not work. I frequently get calls or text while riding and can pull over safely or retrieve after my workout. I have taken calls from the watch and just talked into the device (what other bike computer will let you do all that?). I did disable the lock screen and just tap the screen or push the crown to wake from 70 second sleep (make sure you have the wake screen set to "resume to previous activity" in settings).
Ride safe with your apple watch!