じゅう is 漢音, which is the "received pronunciation" around the capital areas (洛陽、長安) of the 隋 and 唐 dynasties.
にゅう is 呉音, the southern pronunciation of the 呉、越 region during the 六朝 period and before the official contact was established. It is most commonly found in Buddhism-related words.
In Chinese phonology 柔 is a 日母字, whose initial consonant is believed to be */ȵ/ in pre-Qin era, but drifted towards */nʑ/ during Sui/Tang, and ended up "r" or "er" in Modern Standard Mandarin. (The n-initial is somewhat preserved in Wu dialects. We know for example 人 is pronounced like "ning" in Shanghainese.)
These characters (e.g. 日、人、入、肉、然、二、児), if they have entered Japanese twice, then very likely they will have a n-based 呉音 and a z-based 漢音 or 慣用音. For example, 毎日(まいにち) vs 昨日(さくじつ), 三人(さんにん) vs 外国人(がいこくじん), 入手(にゅうしゅ) vs 入水(じゅすい), 小児科(しょうにか) vs 幼児(ようじ). We as language learners will just have to remember which is which I guess.