samui asa ha oyu wo nomimasu.

Because 寒い is an i-adjective, and i-adjectives can directly modify a noun.

• 寒い朝 = a cold morning / cold mornings

You do not need の here.
You also do not change 寒い to 寒く when it directly modifies a noun.

Compare:

• 寒い朝 = a cold morning
• 朝が寒い = the morning is cold
• 寒くなる = become cold

So in this sentence, 寒い is simply describing 朝.

Here, は marks 寒い朝 as the topic.

So the sentence is talking about cold mornings as the context:

• 寒い朝は、お湯を飲みます。
• As for cold mornings, (I) drink hot water.

This is slightly different from just marking the subject.
It gives a feeling like:

• When it’s a cold morning...
• On cold mornings...
• As for cold mornings...

Japanese often uses は to set the scene or establish what the statement is about.

Japanese often leaves out the subject when it is obvious from context.

So instead of saying:

• 私は寒い朝はお湯を飲みます。

Japanese very naturally says:

• 寒い朝はお湯を飲みます。

The listener usually understands that the speaker means I unless another subject is being discussed.

This is extremely common in Japanese. English usually needs a subject, but Japanese often does not.

を marks the direct object of the verb—the thing being drunk.

• お湯を飲みます = drink hot water

So:

• お湯 = hot water
• を = object marker
• 飲みます = drink

Whenever an action is done to something, を is very often the particle you will see.

Examples:

• 水を飲みます = drink water
• パンを食べます = eat bread
• 本を読みます = read a book

飲みます is the polite non-past form of 飲む.

In Japanese, the non-past can mean:

• a habitual action: drink
• a future action: will drink

In this sentence, it most naturally sounds habitual:

• On cold mornings, (I) drink hot water.

So it is not just one specific morning unless context says so. It suggests something the speaker usually does.

Also:

• 飲む = plain form
• 飲みます = polite form

お湯 means hot water or warm water, not plain water in general.

• 水 = water
• お湯 = hot water

The お here is a polite prefix that appears in many everyday words.

Examples:

• お茶 = tea
• お金 = money
• お湯 = hot water

Even though お originally has a polite function, お湯 is just the normal word people use. It does not sound unusually formal.

This is a very common beginner question.

In particles:

• は is pronounced wa
• を is pronounced o

So this sentence is read:

• さむい あさ は おゆ を のみます
• pronounced roughly: samui asa wa oyu o nomimasu

This is a historical spelling pattern that remains in modern Japanese.

So remember:

• normal は inside a word = ha
• particle は = wa
• particle を = o

Yes, Japanese word order is more flexible than English, as long as the particles make the roles clear.

The standard order here is:

• 寒い朝は お湯を 飲みます。

But Japanese could also rearrange parts for emphasis, for example:

• お湯を寒い朝は飲みます。

That said, this sounds less neutral. The original sentence is the most natural basic order.

A useful rule for learners is:

• Put the verb at the end
• Use particles to show each word’s role

These are very different structures.

• 寒い朝はお湯を飲みます。
  = On cold mornings, (I) drink hot water.

Here, 寒い modifies 朝. It means cold morning.

• 朝は寒いです。
  = Mornings are cold. / It is cold in the morning.

Here, 朝 is the topic, and 寒い is the main predicate.

So compare:

• 寒い朝 = a cold morning
• 朝は寒い = mornings are cold

That difference—adjective + noun vs. noun + は + adjective—is very important in Japanese.