It's all a big mystery.
I don’t think that it’s a mystery! I think that one needs to look at the family very logically, without any preconceived thoughts.
1827 Marriage John Baxter and Ann Richards at St Neots
1829 St Neots John & Ann baptised Benjamin & John
1831 St Neots John & Ann baptised William
1838 Harriet Florence b 1838, baptised 1856 St Pancras, parents John & Ann
1841 St Neots Ann 32, Benjamin 13; William 10, Mary 8, David 8, Harriet 6
1850 St Neots Benjamin marries – father John Baxter (not stated to be deceased, which doesn’t mean that he wasn’t dead).
1851 Clerkenwell, London Ann 39, William 19, Harriet 14. All b St Neots
1851 St Neots Benjamin 23 & brother David 16. both b St Neots
1860 Harriet marries – father John Oliver Baxter engineer (not stated to be deceased)
1861 Ann is Ann Welton living in Islington with some of her/her son Benjamin’s Summers relatives from Yelling.
1881 Harriet Florence Cooke 43 b St Neots with mother Ann Welton 70 b St Neots
It’s clear that all the above are the same family – the key person is Benjamin whose baptism links him to John and Ann; who is in the same household as Ann, David and Harriet in 1841, whose 1850 marriage names his father as John, and who appears in 1851 with his brother David.
Notwithstanding the lack of a baptism in St Neots, Harriet’s 1856 baptism names her parents as John and Ann and gives her birth date. Clerkenwell, where she was living in 1851, and St Pancras where she was baptised, are adjoining, so I don’t doubt that it’s the same Harriet. The only record of her father being John
Oliver Baxter is on her marriage certificate. I think this is another of Harriet’s flights of fancy.
What needs to be found are John Baxter’s death/burial, which I don’t think was necessarily pre 1851 (he may have done a runner); Ann’s remarriage to Samuel Welton some time between 1851 and 1861, which I can’t find; then confirmation or otherwise that John Baxter baptised 1807 at St Neots is the right one (in the absence of any other John Baxter baptisms in St Neots I think he's the only game in town, although he could conceivably have been a lot older than Ann.
An Ann Richards was baptised at St Neots on 27 Dec 1807, daughter of William and Mary, victualler
David