Very late to respond to this, but I am also descended from Charlotte Alice Butler and her son William Cullen. My link is through William's daughter, Lottie Fannie Cullen, then her daughter Lottie May Moore, her son Edward Moore Beckett, and his daughter Dorothy Margaret Beckett, who was my mother.
With regard to the original query, I am also looking for this information. I have a note written by my grandmother (wife to Edward Moore Beckett), who copied this from an earlier note which I believe was written by Lottie May Moore.
"Margaret Kitchin, whose people were shipbuilders of Liverpool married 1) Mr Butler, a ship builder, the year the King of France was beheaded (1793)and had one child Alice Charlotte Butler born Apr 9th 1793. Later Margaret Butler, now widowed, went to Ireland as housekeeper to the Duke of Rutland and married Mr Maguire, the butler. They had seven daughters, all called Margaret, who died in infancy. She died in Leamington hospital through burns from a ***** of beeswax and boiling oil boiling over. Her husband died while she was in hospital.
The first daughter, Alice, married Thomas Cullen at St Martin in the Fields in 1816. She had four children. She separated from her husband and left one boy in London and went with her half-sister Eleanor Maguire to Leamington with Charlotte b 1823, James and Margaret and later on William b 1838 at either Leamington or Cheltenham (William later went to Aldburgh and became a pastry cook of note). The children were put out to nurse and mother went as a nursemaid.
Not sure how much of this can be proved...I do have two additional children for Alice, Elizabeth and Charlotte, but both of them died as young children, so that is probably why they were missed out. Alice's sister Eleanor exists as she was a witness at Alice's wedding, but beyond that I've struggled to find anything else to prove the rest of the story.
Did you ever find anything more about Alice, and how are we related, would love to know more if you are still researching.
PS I am very confident that William is not a son of Thomas as per the comments in the thread so if you know more...
