Is there a word or an expression that describes the bad mood caused by bad weather?

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:

1. Dark or poorly lit, especially so as to appear depressing or frightening; and
2. 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 !

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.