Lillian was definitely a nn for
Elizabeth at first: in 1279 Lilion was recorded in England. In 1296 Lylie was recorded as a diminutive of
Elizabeth. (See
http://snipurl.com/a2vp for both, they're the last two names.)
So I don't see why
Lily can't be a nn for
Elizabeth, though personally I think
Lily stands fine enough on its own, since its common use didn't evolve from it being a nn name.
Miranda