After thinking that I had exhausted all of the English records, I decided to do a barebones search for the Anderson family in the 1851 census. I entered the surname Anderson and the lived in place of Ashton Under Lyne. I started picking through the results and I hit the family...with one additional child that I didn't know about.
See England, Lancashire, Ashton-Under-Lyne, Household 108 (Wood Street):
James Anderson (b. 1805 Ireland) is ____ maker...I can't read what kind of maker.
Agnes Anderson (b. 1804 Ireland) is ____ maker...I can't read what kind of maker.
William Anderson (b. 1830 Ireland) is general labourer.
Alexander Anderson (b. 1833 Ireland) is shoe maker.
Agnes Anderson (b. 1834 Ireland) is cotton winder.
Mary Jane Anderson (b. 1837 Ireland) is a cotton winder.
James Anderson finally appears on a census! I've had scant information on him so finding this census showing the family association is a welcome record.
Agnes Anderson is indexed as Mary Anderson; however, when you look at the digitized copy, you can compare Agnes (wife's name) against Agnes (daughter's name) and see that they are the same. The digitized copy is difficult to read because the original is badly faded. Also, Agnes' 1804 birth date opens the possibility that she and her brother Joseph were twins. Twins run in the family mostly for female family members.
William is the "new" son. So far I haven't found much on him. Because he is not listed in his uncle Robert McKee's 1881 intestate letters, I'm working under the assumption that he dies before 1881. Not that my working assumption has done me any good to this point.
Alexander Anderson is noted as being a shoe maker, which is his listed occupation on subsequent censuses.
Agnes and Mary Jane are both there too, working as cotton winders. According to Wikipedia, children started working in Victorian England at age 7. They had to because no one was at home to take care of them and the money they made was necessary for the survival of the family.
Again, this experience reinforces the notion that researchers should track down every last census entry.