Miss A. Batley
I believe I have now identified who A. Batley is and her relationship to 'Margot' Gaye.
I achieved this by reading the English National Probate Calendar, looking at all deaths for BATLEY females with the first initial A, from 1955 and working backwards. That process identified one entry where probate was to Louis Margaret Clayton Gaye widow. The middle name Margaret could be shortened to 'Margot'.
England Probate Indexes
BATLEY Agnes Emma of 52 Latymer-court Hammersmith London W6 spinster died 10 February 1953 Probate London 11 June to Louise Margaret Clayton Gaye widow. Effects 2947 pounds.
Death indexes for England reveal her death registration, supporting the probate index information, and a burial index at Ancestry.com indicates she was buried at Seaborough, Dorset (it would be interesting to see the headstone):
Name: Agnes E Batley
Birth Date: abt 1862
Date of Registration: Mar 1953
Age at Death: 91
Registration district: Hammersmith
Inferred County: London
Buried Seaborough, Dorset
14 Mar 1953, 91 years
Agnes was 91 when she died, making her born about 1862. Searching census records revealed Agnes E Batley living with her family in 1871, which I have summarized here:
1871 - Family in 7 Kensington Park Gardens, Kensington, London. John Batley, head, 47, arms and ammunition manufacturer, born Nolinforth, Yorkshire. Maria L Batley, wife, 37, born Lille north of France. Louisa M., dau, 17, born York, Leeds. Annette N., dau, 16, born York, Leeds. Edith B., dau, 13, born York, Leeds. Blanche M., dau, 11, born York, Leeds. Agnes E., dau, 9, born York, Leeds. Ralph C., dau, 8, born York, Leeds. Plus one visitor and five servants.
Agnes' father was John Batley, an 'arms and ammunition manufacturer', with French wife and a large number of children largely born in Leeds, though in 1871 they resided in Kensington, London. Among Agnes' sisters is an older sister Blance M., who will become relevant shortly in this discussion.
John Batley was a successful manufacturer. I presume that obituaries were published on his death but I have not yet found one, however there is information available on the web here and here. It is likely that there are archives holding Batley correspondence that may be instructive.
Agnes BATLEY's probate was granted to Louis Margaret Clayton GAYE widow, apparently called Margot. The SLNSW statement indicates she was a niece of Miss A. Batley. Her death registration was identified:
Birth Date: abt 1890
Date of Registration: Sep 1968
Age at Death: 78
Registration district: Surrey South Western
Inferred County: Surrey
This means Margot was about 65 when she found Worgan's letter among her mother's possessions and donated it to the SLNSW, and she 78 when she died, so was born about 1890. We know Margot Gaye was married (Mrs), and the marriage indexes for England identified her marriage in 1914, and revealed her maiden name to be RUSSELL. Margot was widowed by 1953 according to the probate entry.
Russell Louise M C Gaye Bedford 3b 742
Gaye Arthur D Russell Bedford 3b 742
Based on the marriage and death entries, Margot's birth registration in 1890 could be found:
Births Sep 1890
Russell Louise Margaret C Bradford, Y. 9b 235
The district Bradford, Y. is an alternative name for Bradford and it spans the boundaries of the counties of West Riding of Yorkshire and West Yorkshire
To confirm the relationship of aunt/niece between Miss A. Batley and Mrs Margot Gaye, it was necessary to identify Margot's presence in the censuses in the hope of finding her parents. Presumably one of the parents would be a BATLEY sibling to Agnes.
1911 - Louise Margaret Russell (20, born Calverly), with family (parents John Clayton and Maude Morris RUSSELL and sister Sibyl. Living Bedford.
So Margot's parents were John Clayton and Maude Morris RUSSEL. There is only one possible match in the England marriage indexes, for 1882, and that is of John to Blance Morris BATLEY, Agnes' sister.
Marriages Mar 1882 (>99%)
Batley Blanche Morris Windsor 2c 608
RUSSELL John Clayton Windsor 2c 608
This work identifies who 'Miss A. Batley' was, and also confirms the relationship between her and 'Mrs. Margot Gaye'.
The next step for me is to investigate the BATLEY family to try and identify a possible connection to the WORGAN family, more specific than obvious factors such as London. It seems unlikely that the Worgan letter passed down through the BATLEY family, given that John BATLEY was from Yorkshire, but there is more to reveal, and more to do!