I had the same problem and no one at Apple neither Netflix was able to help. I tried several options until I had my Netflix service working on Apple TV.
This is what I've done:
I updated to the latest software and after that I was unable to login to Netflix. Whenever I clicked on the Netflix app I just had the spinning wheel running for hours without doing anything else.
I live live in Mexico, and I am English speaking, so the default language in my Apple TV is English.
I switched to Spanish and then tried to access Netflix and for my surprise I was able to login. After I logged in, I went back to Settings and switched back to English and everything is working perfect now. Try this. This might help you all.