You describe the live weather bug from 2nd flight in the same session of FS2020.
Read the first post in the link below: idea: never select live weather in the world map. Select PRESET, with clear skies, then Fly now, then in the flight, get weather, and in the DROP DOWN list, select live weather. If you stop the flight, start a new one and do the same you will have live weather.
Live weather fix (workaround) - Community / General Discussion - Microsoft Flight Simulator Forums