Hello Sarah,
Thank you very much indeed for all that detailed information!
I hope you don't mind if I try to make a timeline for my own benefit ...
Ca 1849, Liverpool Lancashire
William BURNETT b Liverpool Lancashire (place from 1881 census, birthdate from age of 28 at 1877 marriage and age of 32 in 1881 census ), father William BURNETT (father's name from marriage cert).
1855, Mar qtr Portsea Hampshire
Anne Maria Alma COOK born, father William COOK (father's name from her marriage cert).
1871 census 36 Greystone Houses Botchergate Carlisle Cumberland
Annie's parents and siblings - William COOK 42 Labourer Ry Navvy b Carlisle, wife Emma 39 b Portsmouth, children Kate Emma 11 b Portsmouth, William 9 b Halifax Nova Scotia, George Henry 7 b Gosport, Margaret 5 b Plymouth, Jessie 2 b Woolwich.
1877 Dec qtr Carlisle Cumberland
William BURNETT 28, Widower, a Railway Good Guard, father William BURNETT Lieut in the 11th Regiment of Foot, marries Annie COOK 23, Domestic Servant, father William COOK a General Labourer.
1880 Sep qtr Wakefield Yorkshire
William BURNETT jnr born in Normanton. Informant is his mother Annie; father William's occupation is Railway Good Guard
1881
1881 census Normanton, Yorkshire
BURNETT
William, Head, 32 b Liverpool, Raily Good Guard
Annie MA, Wife, b Plymouth
William, Son, 6 months b Normanton
1881 census 39 Greystone Botchergate Carlisle Cumberland
William COOK (Annie's father) Head Married 51 Gen Labourer Chelsea Pensioner is with children Kate Emma 21 Machine Winder, Jessie 12, James 9,
Rose 6 (James and Rose both b Carlisle)
1881 census Hesket in the Forest Cumberland
Emma COOK (Annie's mother) Married 53, George COOK 16, both b Portsmouth are servants in the household of a Farmer, Peter VICKERS
1882 Dec qtr Newcastle T, Northumberland
Violet Emma BURNETT born.
ca 1884 Gateshead, Co Durham
(There's more than one possibility - a James, and a James William in Jun qtr 1884, and a James Ernest in Jun qtr 1885)
James BURNETT born
1886 Jun qtr, Newcastle T, Northumberland
Gladstone BURNETT born
1887
1887 Jun qtr, Carlisle Cumberland
William BURNETT jnr dies on 8 April at 5 Greystone
Informant is his father William BURNETT, Insurance Agent
??3 May 1887??, South Shields
Rose BURNETT born?
Birth entry not found
Her death certificate gives her date and place of birth as 3 May 1887, South Shields; however, her age is given as 2 in the 1891 census and 12 in 1901. On her marriage cert in 1912, her age is given as 24.
1887 Jun quarter Carlisle Cumberland
Rose COOK (Annie's sister) dies aged 13 on 20 June
1890 Jun qtr, S Shields, Co Durham
Lily BURNETT born at Westoe. Informant is Annie; father William's occupation given as Insurance Agent.
1891
1891 census, Westoe, S Shields, Co Durham
Annie BURNETT is listed as Wife, 35, Married, b Portsmouth. With her are children Violet E 8, James 6, Gladstone 4, Rose 2, Lily 10 months (Rose and Lily both b S Shields. No Head of household is recorded.
1891 census, 6 Melrose Tce Botchergate Carlisle Cumberland
COOKE family (Annie's parents and sibling - thanks for reference Jennifer) were William 69 Coal Agent, Emma wife 59, daughter Jessie 22 Woolen Weaver.
1893 Mar qtr, Carlisle Cumberland
Lily BURNETT, age 3, dies at 40 Greystone; death registered by Annie, father recorded as an Insurance Agent.
1901 census, 16 Union St Botchergate Carlisle Cumberland
Annie BURNETT is listed as Wife, 46, Married, Charwoman. With her are children Violet E 18, James 16, Gladstone 14, Rose 12. No Head of household is recorded. (James's occupation was one I've not come across previously - Electric light shade Cleaner.)
1911 census
Not yet available for Cumberland
1912 Dec qtr Carlisle Cumberland
Rose BURNETT marries Arthur GRUBB on 9 October; Rose's age is recorded as 24. Rose's father is listed on her marriage certificate as William BURNETT, Railway Guard, deceased.
JAP
Edited to add information and to amend information which has been revised in later posts