I use the Beautiful Widgets Weather app, but I also have need for an actual weather alert app. I live in a tornado area and we have weather radios, but they have gone down, so I keep one on my phone. It alerts for anything, snow to volcanoes, shelter in place, law enforcement emergencies, etc, depending where you are located at the time.
Also a good one that will show the temp in the status bar, will show a few different radar maps and other storm alerts on the map and will alert in you in the notification screen if you want, is Intellicast. I don't have the temp in the notification bar on mine. It is not a widget type, but has 10 day forecast and even hourly forecasts, in the app and other stuff. I'll do a few screenshots. You can also use GPS location, so it will follow you if you travel.
The alerts on this one aren't loud, so if you need a substitute for a weather radio, this won't work, but the app I mentioned above will. There are also more pages of what this one shows and you can choose, I just can't add more than 5 screenshots.