I have a theory - as is seems that Anthony Bertholini senior was never married to Hannah Caroline (Caroline would have been a Christian name, mothers surname not shown in the parish records) thought I would try and find a Hannah Caroline.
Could only find an Ann Caroline Wheat (no Hannah Caroline's) who married a James Clarke, sugar baker, 13/3/1830 Sculcoates All Saints, Yorkshire. In 1841 they are living at Sutton, Yorkshire but their 4 children at that time were all baptised in Drypool (where Anthony later was).
Whilst I can't find Anthony junior in 1841, I then found in 1851 an Anthony Wheat 23 born Hull, shown as nephew to a Mary Knapp 74 wid. who's dau. Maria Parr 50 widow is also there. I can find no trace of a baptism for an Anthony Wheat nor is there any sign of him after this census. So, was Anthony Wheat actually your Anthony Bertholini junior? After juniors birth, senior did a runner and en route to London met and married someone else whilst Hannah Caroline/Ann Caroline also married herself. Quite how Anthony Wheat is a nephew to Mary Knapp I have no idea but if Mary gave the details to the enumerator, knowing that his parents were not married did she simply give his surname as his mothers maiden name??
All supposition on my part but felt I should mention it.
Annette