I now have a copy of Jessie's birth registration which shows a dob of 15 October 1889 which matches that shown on the 1939 Register so all seems right.
The birth was registered in Kensington as suggested and it shows her being born at Kensington Infirmary.
She is named as Jessie daughter of Esther Garrett. domestic Servant of 23 Park Street with no father's name given

.
So - thanks to all your suggestions I now have confirmation of my grandmother's birth. Now to check out the marriage of Fred Etherington to Esther Garrett to verify her father's name and occupation.
Having looked on the GRO site I was able to order a PDF of Jessie's birth certificate but when trying the same option for the marriage certificate it does not offer the PDF version as an option which is strange.
Does anyone know if there is likely to be an image or transcription of the marriage certificate online anywhere? It took place in Windsor Sept Qtr 1893 their names being shown as Fred Etherington and Esther Jane Garrett.
Many thanks everyone!