"I met my childhood friend after a long time"
오랜만에 어린 시절 친구를 만났어요.
오랜만에 어렸을때 친구를 만났어요. (어렸을때 -> 어렸을 때)
오랜만에 어릴 적 친구를 만났어요.
All three are correct and natural sounding.
때 and 적 are interchangeable synonyms, and time in the past can be referred to with both ~ㄹ 때 and ~었을 때.
어린 시절(의) is a way of saying 어릴 때 using a set noun phrase, like "from my childhood" as opposed to "from when I was small". So 어린 instead of 어릴/어렸을 is correct in this case.
So all three are saying the same thing correctly.