Dear Trish,
It was wonderful news that you had been able to find Francis and Elizabeth in the 1841 census. I had looked, but the spelling had me confused. This family tallies with the 1851 census, except that Elizabeth was the head by then. Following are the 1851 results:
Elizabeth Mayell b. abt 1803 Bradford Head Pauper, formerly washerwoman.
Caroline Mayell b. abt 1836(

) Bradford Daughter Pauper, formerly washerwoman.
Ellen Mayell b. abt 1840 Bradford Daughter Scholar.
Hanny (Fanny?) Mayell b. abt 1832 Bradford Daughter House servant.
Hudrick Mayell b. abt 1850 Bradford Son. (What a name!!!)
John Mayell b. abt. 1843 Bradford Son.
Lousa Mayell b. abt 1836 Bradford Daughter House servant. (Probably Louisa)
William Mayell b. abt 1842 Bradford Son Scholar (my great grandfather)
Eventhough there are some differences with 1841, it seems to me that this is the same family. There have been age discrepancies all the way through. 1861 has Elizabeth born about 1793, which would really be impossible.
Thanks, Trish, for all your help. The jigsaw is coming together. I find it really exciting. Best wishes, Nibea.