Not a website, but for quite some time I used an add-in for Firefox, ForecastFox, which effectively showed weather information in the status bar/toolbar. It uses Accuweather.com via an API I believe.
There are also ways to do it with [ulr=https://blogs.technet.microsoft.com/heyscriptingguy/2010/11/07/use-powershell-to-retrieve-a-weather-forecast/]Powershell[/url].
of course as I mentioned these aren't websites, but I think they also sidestep the issue with ads and such, as well as loading fast, since they use the APIs which only really give weather information, rather than stylesheets and formatting and images and such as with the standard websites.