Hello, avm will be able to explain how she connected the Fullers with the Heathers
I was niggled by various things:
The lack of Heather birth registrations for the children other than Louis
The lack of a marriage for Thomas & Kate in the expected names
The lack of a Worthing birth/1871 census entry for a Thomas Heather of the right sort of age
...so on a hunch that another surname might be involved I went hunting for children's birth registrations matching the first name/middle name combinations but leaving out a surname. Finding Fuller births matching time, place and given names for Edith & Emily, I then looked for their baptisms as Fuller, and when they turned up with the 20 Flood St address and father's occupation carman (matching Thomas Heather's address and occupation in the censuses) it all started to fall into place.
As groom says, it would have stayed a hunch had there not been such good London records.
Generally, it is always essential to try to trace birth (and marriage) registrations where they are available. We often see on here people taking an approximate birthdate from a census entry, or an approximate marriage date based on the 1911 census, and leaving it there, but finding those civil registrations can be the key to cracking a puzzle.