First rule of Family History - NEVER ASSUME ANYTHING!
Would be worth getting the marriage cert to see who Hannah names as her father. If John, a farmer, then we are on the right lines.
Certainly it looks like he is he Dad from all the evidence so far.
Second rule - take it one step at a time!
Find out all you can about one person before leaping back to the next, or you may find you are leaping in the wrong direction!
That is why Geof was looking for Hannah and Joseph's children after they settled in Tydd St Giles. And as he said, no christening might (or might not) indicate that the family were Catholic or non conformist .... which might also be true of the Coopers, who knows!
Although Hannah says born Long Sutton, she could have been christened anywhere, so it will be hard to find without knowing a bit more about the family anyway.
There seems to be a possible sister Keturah, who may ne easier to trace. We know she married Collyer. Perhaps we can find her on later censuses to see where she was born - that might give a clue as to whether John was in Long Sutton for any length of time.
All sorts of possible scenarios, but lots of work needed to work methodically through the families movements, if we can.
Hannah gives Long Sutton as p o b on 1851 - do you have her on later censuses? Where does she say then?