Hi Lewis,
Interesting problem.

Seems to me that John and Martha were siblings or more likely half siblings though how you will prove this when they would have been trying to keep it secret I don't know. Have you obtained the birth certificate of all people involved? Also check the baptism records as they sometimes have different information.
If any of the children emigrated then their marriage or death records may give some extra information about their mother though it is likely they would have not been aware of the truth.

The 1901 census has a niece Muriel BRADLEY age 1 and this is probably her birth:
Births Mar 1900
Bradley Muriel Northallerton 9d 693
and death
Deaths Mar 1902
Bradley Muriel 2 Easington 10a 259

Have you obtained her birth certificate and death certificate? It is likely that Muriel is the daughter of the supposed sister Martha Ann BRADLEY shown in the 1901 census or possibly daughter of the sister Jane BRADLEY age 3 in the 1881 census. The death may have been registered by John or Martha and did they say they were father or mother?

I tried to trace Jane BRADLEY after 1881 but think this might be her death:
Deaths Sep 1882
Bradley Jane 4 Northallerton 9d 443

Have you traced the mother Mary BRADLEY who was age 34 in 1881 and unmarried though she had at least two children at that time and then possibly the Martha Ann BRADLEY born Dec qtr 1881?

Even if John and Martha were born to different fathers they could not marry legally and so pretended to be so.

I had thought that maybe the mother Mary Bradley had married some chap called NEESON but I see in 1891 census Mary is still unmarried and living with daughter Martha in the Workhouse in Northallerton. Says Mary is age 40 and Martha age 10. Mary born Brompton and Martha born Northallerton.

I can't find Mary Bradley after 1891 and this might be her death:
Deaths Dec 1899
Bradley Mary 51 Northallerton 9d 447

Bryan