Going back to the original question: I found a school record for a Lily which may or may not be relevant:
Lily McPherson born 7.6.1885* was admitted to school** 7.6.1898. Previous school: Dr Barnardo's.
Father: Fredk., of 12 Spital Square.
* I suggest you take the "precise" birthdate with a pinch of salt. A surprising number of children in the register appear to have had a birthday on the day of their admission. It may be that the school was told the child was (say) 13, without more, so they entered them in the register as though they were exactly 13 that day.
**Unfortunately the name of the admitting school isn't included in the details.