2023 · Weston Weather Forecasts. Weather Underground provides local & long-range weather forecasts, weatherreports, maps & tropical weather conditions for the Weston area. Weston, WV Weather Forecast | AccuWeather Daily Monthly Current Weather 6:34 AM 54° F RealFeel® 53° Air Quality Excellent Wind NNE 3 mph Wind Gusts 8 mph Partly cloudy More Details.