"Gloomy" comes to my mind naturally for God knows what reason. I'm not sure it's appropriate or not. Maybe some even better words or expressions?
No better word than gloomy comes to (my) mind. It just may be you answered your own question already!
"Gloomy weather" is a ubiquitous expression so it's no surprise that it entered your mind so readily.
It is also quite appropriate when you consider its definitions, which Oxford Dictionaries Online gives as:
- Dark or poorly lit, especially so as to appear depressing or frightening; and
- Causing or feeling depression or despondency.
The first definition is perfect for dark, depressing weather, i.e. gloomy weather, which in turn causes people to feel depressed, i.e. gloomy !