I set this up today too, and continued to get that 401 error even after confirming email and verifying API keys etc, until I set up the Billing Name / Address / Payment Method (I chose PayPal) within OpenWeatherMap by clicking the Subscribe button -- I immediately also set the limit to 1000 API calls a day (it defaults to 2000), so I will never be charged for it.
Once I did that "Subscribe" (even though its going to be free), the 401 errors went away and I got data -- both in my web browser when calling the URLs directly with my API key, as well as within the Nodeserver.
residualimages's post in OpenWxMap: Connected but no data after install was marked as the answer