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Thanks for this and the earlier comment. Martha's surmame is given as "Kennell, formerly Bunting" on the 1840 birth certificate of a daughter; I did wonder whether Bunting was her maiden surname but her baptism in 1825 says her parents are Joseph and Mary Bunting. The Piddington marriage register doesn't help as it doesn't say whether Martha was a spinster.
I will see if there is anything more to be unearthed about Joseph (1260) and Mary; I suppose it depends on finding ot whether they had further children other than those currently recorded on KEN New Romney. If they did move to Hackleton or Piddington, then the reason is not obvious.
The Northants Record Office does have a register containing some details of the early Baptist congregation at Long Buckby, though it is far from complete. I didn't know of its existence until quite recently. It does however mention Martha's baptism and death in one of the membership lists.