My classmates had advanced from an affiliated elementary school without having to pass an entrance exam.
こんにちは!
ご質問ありがとうございます。
ご質問について、いくつか言い方が考えられますが、例えば、
I attended a private middle school where some of my classmates had advanced from an affiliated elementary school without having to pass an entrance exam. In comparison to me, who attended a public elementary school, those classmates possessed a significantly higher level of English proficiency as they had more time to dedicate to English lessons during their earlier years.
として、ご質問にあるような内容を伝えるのはいかがでしょうか。『[エスカレータ式](https://eikaiwa.dmm.com/uknow/questions/11681/)』という制度があまりない国や地域もありますので、そのままではなく、advanced from an affiliated elementary school without having to pass an entrance exam というように、『同じ系列の小学校から試験に合格する必要なく進学したクラスメイト』というように表現しています。
参考になれば幸いです。