Hi Clare,
A tough one, isn't it...
You may have already checked this, but the 1891 census (6 months before Alice's birth) shows Emma OAKHAM (+7 children) living at 24 Church Path, Fulham - RG12/46, f 123, p 11
Probably no relation, but you never know.
Also I noticed that the electoral rolls give Harry as Henry Reed Nelson - he may therefore appear in the missing censuses as Henry.
Sorry, not much help, will keep looking.
Alexander