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I am not able to understand the following use:

~がきにならなくなっていった

Does it mean I am/am not interested about the subject?
Can someone explain the conjugation used here please?

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「~~が気{き}にならなくなっていった」

「~~が気に + ならない + なる + いく」  

The last part いった is from いく(行く) and not from いる like you stated in your comment above. I am sure you know both いく and くる are often used as subsidiary verbs. (The past tense form of いる is いた, and not いった.)

「気にならない」 means "to not be concerned", "to be free from", etc.

「なっていく」 means "to (gradually) become". It expresses a shift from one state to the next.

Old state: Concerned

New state: Unconcerned

Thus, 「~~が気にならなくなっていった」 means:

"(Someone) became less and less concerned about ~~ (over time)"

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