Hi
The nearest death registration on FreeBMD with a birth date of circa 1855 would be the one given below. Hallam Street is in Paddington so this death registration would be consistent with the 1916 address
Deaths Jun 1936
Walker Eliza M 81 Paddington 1a 77
http://freebmd.rootsweb.com/cgi/search.plThe transcription of the General Registry Office index for BMDs does not have complete coverage of the 1930s for deaths
http://freebmd.rootsweb.com/progressD.shtml#1930A death notice may have been inserted in a local newspaper which would give an indication of where she was buried. In the case of Paddington that would be Westminster Archives (their newspaper collection may be wider than just Westminster)
http://www.westminster.gov.uk/services/libraries/archives/They also hold the registers for Hanwell cemetery (not sure up to what date)
In the case of Streatham that is Lambeth Archives (they also hold copies of the burial registers - but not the further registers held at the cemetery office if the grave was purchased which give further details on who purchased the grave and who was buried and when at what depth and whether a gravestone was purchased). They will also hold local newspapers.
http://www.lambeth.gov.uk/Services/LeisureCulture/LocalHistory/Archives.htmCopies of wills can be obtained from
London Probate Department
PRFD, First Avenue House
42-49 High Holborn
7th Floor
Holborn London
http://www.hmcourts-service.gov.uk/cms/1176.htmIt might be worth obtaining a copy of Eliza Marie's mother's will
Sarah Maria Eveleigh died 22nd June 1906
It might give some clue about her daughter during the missing years.
On the 1901 census Janet Kathleen and Marie Gertrude Walker were scholars at Ackworth school which is a very famous Quaker school which would indicate the family were Quakers? Ackworth school's records may give further information
http://www.nationalarchives.gov.uk/nra/searches/subjectView.asp?ID=O37607http://www.archives.wyjs.org.uk/wyjs-archives-wakefield.aspIf the family were Quakers then the Society of Friends library in London might be useful as the Quakers were copious record keepers
http://www.quaker.org.uk/libraryThe first two baptisms for the children George Hunter Eveleigh Walker (1880) and William Fennell Walker (1883) at St Leonard Streatham indicate the family were not Quakers at that time.
Regards
Valda