Think we are making some headway on who Gertrude was. Made me chuckle, mshrmh, that it brought me full circle to your post here:
I couldn't find a William Ayton on the 1891 for England & Wales who looked as though he'd be described as a translator. The census was 5 April, so just before the date given for Gertrude's birth on the death registration, assuming that is her & the date was accurate.
Clutching at straws: on the Historical Directories site the 1895 Slaters for Manchester & Salford records a Jonas Ayton at 49 Perth Street, Cheetham. He was a compositor. At the same address on the 1891 census is James Ayton with his wife - he's born Scotland. I suspect this is really Jonas, as LancashireBMD has a marriage in 1888 at Salford that matches the couple. Jonas's baptism could be the one on FamilySearch in 1864 Edinburgh - parents William Ayton & Margaret Taylor. The same site records the baptisms of other children to the same couple - William Taylor (1865), Nimmo Crawford, Christina, James & William (1874).
Coming at this in a different way, in summary, I think that William Ayton and Margaret Taylor that you mention were indeed Gertrude's grandparents. However, parent not James I think but eldest daughter Christina b. 1868 in Edinburgh. Can't see her easily in 1891 but everything points to her marrying a William Jacob in Liverpool in 1892 (explains why daughter Gertrude Margaret was registered under Ayton).
I think this is the family back in Edinburgh with young Gertrude for 1901:
William Ayton 63, Printer's Rader (printer’s reader….could translator of foreign languages be a polite version of the same thing?) b. Edinburgh
Margaret Taylor 58 b. Keith, Banff
Robert Ayton 24
Margaret Ayton 21
George Ayton 20
Annie Ayton 18
Gertrude Jacobs 9 – granddaughter b. EnglandAddress: 48 Rosemount Bdgs, Edinburgh
From playing around with the index on Scotlands People for 1911, Gertrude showing now as Ayton looks to be still living with her grandparents.
I think this is her mother and (father?) William Jacobs down in Hackney, London for 1901 (RG13/225/74/3):
William Jacob 45 [struggle with his occupation?] b. Newcastle
Christina Jacob 33 wife b. Edinburgh
I think from what we have so far, Gertrude reverted to Ayton in later years. Also, I think to cover the fact she was illegitimate, she likely gave her grandfather William Ayton's details as those of her father on her 1916 marriage registration.
...And then I am stuck again

Monica