You could always call a child whatever you want as a nickname, but structurally it doesn't work. No one would really get it, unfortunately. But like I said, nicknames are anything goes. Yeah, it's a tricky name for nicknames...I'm stumped.
Why not? If Lola can work as a nn for Viola, then why can't Lily work for Cecily? I do agree it's a bit contrived (I think Viola nn Lola is a bit contrived), but it works. Why not. I'm not the hugest Cece fan, either, but I do like Celie (SEE-lee, sort of like Seeley), and I think Cessy (to rhyme with Tessie) is pretty cute, too, or even Cissy. I read a novella whose main character was Cecelia, nn Cissy, and that always stuck with me. I don't like the idea of Sissy as a nn, but somehow, Cissy changes it; I think it's rather cute.