It was not until the Legitimacy Act of 1926 that children born before a marriage could be subsequently legitimised by a marriage between their parents.
I have examples of births which were illegitimate which were registered as legitimate. Just a matter of bluffing it out with the registrar. I also have examples of late marriages, some late for no real reason, they just didn't get around to it (some just didn't get around to it at all). Of course the main reason in this case for the late marriage would be a previous marriage by either partner where the spouse was still alive.
The most obvious way to test whether the mother was the same woman is to compare the 1861 census details with the 1851 census information. From your details I think this is the family on the 1861 census
RG9 2883 folio 32 73 Jackson Street Chorlton upon Medlock Lancashire
Henry Caudwell 44 Stone, Staffordshire, Head Married Joiner
Ann Caudwell 48 Newsome, Yorkshire, Wife Married
Edward Caudwell 15 Eccles, Lancashire, Son (not sure of occupation difficult to read)
Mary Caudwell 11 Mancr, Lancashire, Daughter
John Caudwell 9 Mancr, Lancashire, Son
Sarah Caudwell 7 Mancr, Lancashire, Daughter
If you hold the birth certificates for Mary and John or at least one of them, you can do an area search in Manchester in the 1851 census for the family. You may also find further information from this census - always a possibility of Kaye children which could also be extremely useful.
Regards
Valda