And a key mystery is where most of the children were in 1841, if I'm right about the list above. William was probably the only one already married; Jane, Edward and Margaret were not, and the others probably weren't (based on children's ages in later censuses). And yet only Mary, Margaret and Isabella were with their parents.
The children born pre 1821 would be likely working away from home in 1841.
Trish
Here's my so-far on the locations of William and his siblings in the 1841 census:
William - at Upsettlington, with wife Jane and 3 children. Evidence of connection: William's obituary (Berwickshire News and General Advertiser 22 January 1901) says he "entered, in 1837, the service of the late... Lord Marjoribanks of Ladykirk, and continued on this estate until the end of his working days".
John - died 1838. Evidence: Newspaper item in this thread.
Thomas - at Licker, Lowick. With Matthewson family. Evidence of connection: Thomas and Eliza's eldest child born around 1843; a Thomas Burns married an Eliza Matthewson in 1843 at Glendale.
Jane - at Ravensdowne, Berwick. With Willoughby family. Evidence of connection: Jane Heslop's obituary (Newcastle Journal 1918 Mar 26) says "in her teens [she] entered the service of Mr Willoughby, solicitor, Berwick"; 1851 census shows same William Willoby (same name, age, location) as "attorney in law".
Martha - not found/confirmed. In 1851 census, married to Robert Percy, in Easington, Durham, oldest child age 5.
Edward - at Simprin, Swinton. With Wilson family. Evidence of connection: brother William's obituary (Berwickshire News and General Advertiser 22 January 1901) says William "was for some time at Simprim, under Mr Wilson" [prior to 1837].
Mary,
Margaret,
Isabella - with parents at New Hall, Fowberry, Chatton.