Name from “Turner” by James Hamilton.

Occupation from son’s baptism records.

Parish records of St. Mary Magdalene Bermondsey, Southwark, born April 27, 1818, baptism, May 22 1818, Thomas James Thompson, parents Thomas and Mary Ann, address Bermondsey New Road, father’s occupation is smith.

In the 1861 census 10 Walton Street, Chelsea, Middlesex
Head Thomas J. Thompson, age 42, tutor, born Bermondsey, Surrey
Wife Eliza Thompson, age 40, born Trull, Somersetshire
Daughter Mary Thompson, age 5, scholar, born Pimlico, Middlesex

Other sources say her mother was Hannah Danby, niece of Sarah Danby.

London Marriage Bonds and Allegations
Commissary of Surrey
June 15, 1840
Appeared personally Thomas James Thompson of St. Mary Magdalen Bermondsey in the County of Surrey Bachelor of the age of twenty one years
and prayed a Licence for the Solemnization of Matrimony in the Parish Church of St. Mary Magdalen Bermondsey aforesaid between him and
Georgina Danby of St. Giles Camberwell in The County of Surrey Spinster of the age of twenty one years
and made an Oath, that he believeth that there is no Impediment of Kindred or Alliance, or by any other lawful Cause, nor any Suit commenced in any Ecclesiastical Court, to bar or hinder the proceeding of the said Matrimony, according to the Tenor of such Licence. And he further made Oath, that he the said Thomas James Thompson hath had his usual Place of Abode within the said Parish of St. Mary Magdalen Bermondsey for the space of Fifteen Days last past
signed Thomas James Thompson

If this marriage is correct then there are some “inconsistencies” in the parish records.
Parish church of Bermondsey, Surrey
June 18, 1840
Thomas James Thompson of full age Bachelor, Chemist, Bermondsey Ln?, father Thomas Thompson, dead.
Georgiana Danby of full age, spinster, Camberwell, father George Danby, dead.
by Jno. E. Gibson, rector, in the presence of William Harris, Elizabeth Thompson and Sarah Thompson

In the 1841 census No. 10 Webber Row, London Road, St George the Martyr, Surrey.
Thomas Thompson 20, clerk, born in county.
Georgiana Thompson, age 25, born in county.
At the same address are Mary and Katherine Ferris, ages 47 and 9, and William and Margarette Asbury ages 36 and 35. [This address seems to be close to Waterloo Station]

From Lisa de Garston’s Family Tree
Died February 8 1843, Lying-In Hospital, York Road Lambeth, died of puerperal fever - no surviving child. [The no surviving child seems to contradict her son’s burial in July] [York Road passes by Waterloo Station]

In the January to March 1843 death index Georgiana Thompson, Lambeth, London, volume 4, page 227.

Parish records St. Mary Magdalen, Bermondsey, Southwark, Surrey
February 12, 1843 burial Georgiana Thompson, age 31. Performed by Jno. E. Gibson.

Georgiana was not mentioned in the will of Hannah Danby’s in December of 1853 yet her sister Evelina was mentioned. It would seem to be a reasonable conclusion Georgiana must have died prior to that date.

It seems the probable timeline for this family is as follows, short and tragic.
June 18, 1840 married
June 21, 1841 son Thomas William Thompson is born
November, 1842 son Thomas William Thompson dies
February 1843 mother Georgiana dies at about the same time she gives birth to a second son Thomas Markham Thompson
July 1843 son Thomas Markham Thompson dies

Findagrave.com
Georgiana Danby Thompson
Birth 1811, Surrey, England
Death Feb 1843 (aged 31–32), Lambeth, London Borough of Lambeth, Greater London, England
Burial St. Mary Magdalene
Bermondsey, London Borough of Southwark, Greater London, England
Memorial ID 236611738
Georgiana Turner was the younger of two illegitimate daughters born to the artist, Joseph Mallord William Turner, and Mrs. Sarah Danby. She was married 18 Jun 1840 to Thomas James Thompson at Saint Mary Magdalene Church in Bermondsey. She died following the birth of a son, Thomas Markham Thompson, of puerperal fever, more commonly known as childbed fever.

Second wife.
Name, birth year and marriage date from The Danbys by Selby Whittingham.