OpenWeatherMap provides a free API which can be called from the Browser to get weather information worldwide.
The free tier of the API does not allow access via HTTPS. Unfortunately, now that browsers have tightened
up access restrictions this means that any web app which uses HTTPS can't access the API.
This code provides a simple Java App which runs on Google App Engine and relays requests.
So, if you have a request like
http://api.openweathermap.org/data/2.5/weather?q=Glasgow&APPID=<token>
you can install this relay (for free) on App Engine and call
https://<your-app-id>.appspot.com/data/2.5/weather?q=Glasgow&APPID=<token>
There is a copy of the app running at
https://openweathermap-https-relay.appspot.com
Feel free to use this copy rather than installing your own, but it may stop working if the (very limited) resources are exhausted.
The License is GNU.