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.