One cannot combine 「[慣]{な}れる」 with 「なる」.
「[寒]{さむ}くなる」 is fine because 「寒い/寒く」 is an adjective. 「慣れる」 is a verb.
To express "to get accustomed to", you can say:
「慣れてくる」 and 「慣れていく」.
No other forms could express the "get" part of "to get accustomed to" nearly as closely as these two.
「[大学生活]{だいがくせいかつ}にも慣れてきました。」 = "I am getting used to my college life."
「まだ大学生活には慣れていませんが、これから少しずつ慣れていくと思います。」 = "I am not used to college life yet, but I think I will get used to it little by little from now on."
Notice neither いく nor くる is used in the first part of my second example sentence because it says "I am not used to".