I heard from my mother that my maternal grandparents' family wasere recognized as 富农（fù nóng, rich peasant）after the establishment of New Chinathe PRC.

Note:

If a family was recognized as “地主” (dì zhǔ, landlord) or “富农”, the family's members would be the targets of attacks since New China was supposed to be the country of those who were recognized as “贫农”（pín nóng, poor peasants）.

In the previous example, English people would probably specify PRC as the important detail was the change in government and so the distinction between the PRC and previous iterations of China is important. But since the PRC is the current China, after that point, it would be more natural to just say China.

They live in the same city asthat my uncle used to live in.

My younger aunt got married to a military man and went to the Xinjiang Uygur Autonomous Region with him.

My cousins live in different places, and the distances between the places are so greatvery large.

"so great" leads the reader to expect "that" after it. I think "very large" would be more natural here.

I heard from my mother that my maternal grandparents' family was recognized as 富农（fù nóng, rich peasants）after the establishment of New Chinathe People's Republic of China (PRC).

NOTE: The expression "New China" is a phrase that China experts will understand. However, over the last 5,000 years it seems there have been many different "New Chinas" created over history . . . I recommend the revision about.

If a family was recognized as “地主” (dì zhǔ, landlords) or “富农”, the family members would be the targets of attack since New Chinathe PRC was supposed to be the country of those who were recognized as “贫农”（pín nóng, poor peasants）.

My younger aunt got married to a military man and went to the Xinjiang Uygur Autonomous Region with him.

也有可能: My parents have both diedpassed away.

注：“passed away ”比 “died ”听起来更柔和。

My faSUGGESTION: Specify the year of their died earliereaths; it sounds smoother.¶
EXAMPLE: My father died in 2001 while my mother died later.in 2012.

I have rarely have a chance to get in touch with my cousins after the passing of the oldprevious generation.

I heard from my mother that my maternal grandparents' family was recognized as 富农（ (fù nóng, rich peasant）) after the establishment of Newthe People’s Republic of China.

“New China” isn’t a standard term in English. Do you mean the PRC?

If a family was recognized as “地主” (dì zhǔ, landlord) or “富农”, the family members would be the targets of attack since New China was supposed to be thea country of those who were recognized as “贫农”（ (pín nóng, poor peasants）).

More natural.

My eldest uncle got lost in a war, and no one knew (or: knows) what happened to him.

My younger uncle left home to go to a coal city in Liaoning Province and worked there, got married there, and lived there until he died.

It sounds better if you don’t repeat “there.”

He hasd two kids: onea son and onea daughter, and both of them are {of a similar age to me | around my age}.

“Had” (past tense) because he’s dead.

My younger aunt got married to a military man and went to Xinjiang Uygur Autonomous Region with him.

More natural.

She hasd three children: two sons and onea daughter.

My younger uncle died one or two years after my aunt's death, and his wife is still alive.

“My uncle” is confusing, because you just mentioned your aunt’s husband, and your aunt’s husband is your uncle, too. We have one word, “uncle,” for 舅舅, 姨丈, 叔叔 and 伯伯.

My father died earlier while my mother died later.

alternatives:
My father died before my mother.
My father predeceased my mother. (formal)

My cousins live in different places, and the distances between the places are sovery great.

I rarely have a chance to get in touch with my cousins after the passing of the older generation.

He has two kids: onea son and onea daughter, and both of them are of a similar age to me.

She has three children: two sons and onea daughter.

I never had/ got a chance to see my maternal grandparents since they passed away before I was born.

I heard from my mother that my maternal grandparents' family was recognized as 富农（fù nóng, rich peasant）) after the establishment of New China.

If a family was recognized as “地主” (dì zhǔ, landlord) or “富农”, the family members would be the targets of attacks, since New China was supposed to be the country of those who were recognized as “贫农”（pín nóng, poor peasants）).

My mother had three siblings: my two uncles and one aunt.

My eldest uncle got lost in a war, and no one knew/ knows what happened to him.

My younger uncle left home to go to a coal city in Liaoning Province and worked there, got married there, and lived there until he died.

He has two kids: one son and one daughter, and both of them are of a similar age toas me.

My aunt died about ten years ago, and/ but her husband is still alive.

My uncle died one or two years after my aunt's death, and his wife is still alive.

My father died earlier, while my mother died later.

My cousins live in different places, and the distances between the places are so gream is so great/ far apart.

I rarely have a chance to get in touch with my cousins after the passing of the old/ older generation.