It's not that 時 is read as し and 雨 is read as ぐれ, it's that there was a Japanese word, しぐれ, and this pair of kanji together were chosen to represent the full word together. It can't be split and divvied up to either individual kanji.
This is a fairly common phenomenon, and some basic words you probably know are part of the same category, which is called 熟字訓 (jukujikun).
For instance, きょう, written in kanji as 今日, or おとな, written in kanji as 大人. These are the same kind of thing, where you can't break up the readings to the individual kanji, they are readings that are assigned to the pair together.
Wow. Cool! I've been studying for years and I've never heard of this... I guess I just never questioned it until I heard TK. I relly admire your knowledge. Thanks!
Why is this reading so different?
I have wondered this since the first time I encountered the band 凛として時雨. Why is it read as しぐれ and not something more like ときあめ? I guess I can understand 時, as its 音読み is ジ. But in no other situation (as far as I have seen) is 雨 read グレ. Please help me...