Learn Hardcore Japanese: asita no tenki ha kumori desu.

の here is the genitive (possessive) or “of” particle. It links 明日 (tomorrow) to 天気 (weather), so 明日の天気 literally means “the weather of tomorrow,” i.e. “tomorrow’s weather.”

What role does the particle は play in 天気は曇りです?

は marks the topic of the sentence. It tells the listener, “I’m talking about 天気 (weather).”

Topics can be thought of as “what the sentence is about.”
You’ll often see は instead of が when the information is already known or when you want to comment on the topic.

Why is は pronounced “wa” instead of “ha” here?

Although は is spelled with the “ha” hiragana, when used as the topic particle it is always pronounced “wa.” This is a special historical pronunciation rule for this particle.

What function does です serve at the end of 曇りです?

です is the polite copula (linking verb) in Japanese. It:

Equates the topic with the descriptor: “The weather is cloudy.”
Adds politeness.
Without です, the sentence would be more casual: 明日の天気は曇り.

Why is 曇り a noun here rather than an adjective?

曇り (くもり) is the noun form meaning “cloudiness” or “cloudy weather.” When you want to say “It’s cloudy” in a polite, simple way, you use a noun + です.
Alternatively, you can use the adjectival verb form:

曇っています (くもっています) – “It is cloudy” (progressive adjective-like form).

How do you read 明日 in this sentence? Are there alternatives?

In everyday speech 明日 is most often read あした.

A slightly more formal/poetic reading is あす (often seen in weather forecasts).
Both mean “tomorrow,” but あした is more common in conversation.

Is the word order in Japanese the same as in English?

Japanese generally follows Subject–Object–Verb (SOV), but copular sentences with です are more like Topic–Comment. Here:

Topic (明日の天気は) → Comment (曇りです)
In English we have Subject–Verb–Complement: “Tomorrow’s weather (S) is (V) cloudy (C).”
The key is: topic goes first, verb (or copula) goes at the end.