There was a theory that another descendent had that 'Kate' was actually Jane - and there are two Janes in the Richards family of Melksham that were cousins and born a couple of years apart, so would make sense if one of them had a nickname so as to avoid confusion. So I'm starting to think that the Jane living in Chelsea in the 1871 census that I originally thought was Florence's mum was actually her mum's cousin.
As Florence is the niece of Henry Richards, do you know his brother/s and families?
There is a marriage that might be worth looking at.
Free BMD
September 1870 London City 1c pg 159
John Adams and Jane Richards
It seems bizarre though with John being a middle name and Jane using Kate
