A few more records to throw into the pot, though I'm not sure that they are all necessarily related.
Some baptisms for children of a John and Sarah HEARSON:
5 Mar 1819 Henry (at Stedham), father - millwright
10 Mar 1827 Louisa Mary (at St Mary, Islington), father- carpenter
8 April 1832 Alexander Thompson (at West Hackney), father - carpenter. Buried 27 Dec 1837 at St John, Westminster)
(b 12 Aug 1834) bap 19 Feb 1852 Sarah Caroline (at St John, Clapham), father - engineer
There is also (b12 Dec 1824) bap 2 Jan 1825 John HOURSON (at St Mary,Rotherhithe), son of John and Sarah - though I can only find a transcription, not the actual image.
When Louisa Mary HEARSON marries in 1853 (to Edward ELLIS), she says her father John is an engineer. Sarah Caroline HEARSON is a witness.
There is a HEARSON family in Westminster in 1841:
John HEARSON 40 carpenter (not born in county)
Sarah 40 (Y)
Louisa 12
Thomas 10
Sarah 6
Frederick 2
HO 107/ 737/16 pg 22
There is a Frederick Louis HEARSON birth registered Dec qtr 1838 in Westminster (1/325) mmn BLAIN.
So this looks like the likely family baptising those children in London, with father John being a carpenter/engineer. Also a burial for a Sarah HEARSON on 8 Nov 1843, aged 45 at St John, Westminster.
So ... I'm not sure that the baptism of Henry at Stedham is the same family who appear in London. The father John remarries in 1848 to Caroline TAYLOR and says he is an engine manufacturer, so that fits with the London family. But Sarah's maiden name was BLAIN, not ANDERSON.
Will think about this some more.