Is this the family in the 1841 at Lime St, All Saints Northumberland?
William Bell 40 (copperas boiler)
Eleanor Bell 40
Jane Bell 21
Sarah Bell 13
William Bell 5
H0107/846/1 pg 21
Only Sarah and son William are indicated as being born in County, and Eleanor and Jane have ticks in the column "born Scotland, Ireland or foreign parts" (as do some others on the page). I do note though that the 1851 says they were born Hexham.
I can see these baptisms for the couple (to see whether the naming helps with Eleanor's possible parents):
Jane 27 Aug 1820 Corbridge (of Prospect Hill, father: husbandman, note: "From Bishop's Transcript - Diocese of Durham."
Thomas 7 April 1822 Corbridge (of Prospect Hill, father: husbandman, note: "From Bishop's Transcript - Diocese of Durham."
William 2 Oct 1825 Longbenton (of Walker, father: labourer) - can't see a suitable burial?
Sarah 17 Mar 1829 Newcastle upon Tyne (of Ouseburn, father: Copperas maker)
Eleanor 9 Nov 1830 Newcastle upon Tyne (of Ouseburn, father: Copperas manufacturer)
William 21 Feb 1835 Newcastle upon Tyne (of Ouseburn, father: Copperas Maker)