"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?
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Gloomy is a good thought here, as evidenced by the fact that it is often applied to the weather (i.e. gloomy weather) as well as the mood caused by such weather. There's a good reason for naturally thinking of gloomy; it is common in this context.
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:
The first definition is perfect for dark, depressing weather, i.e. gloomy weather, which in turn causes people to feel depressed, i.e. gloomy !