You most likely have this but
Samuel was buried 14/09/1851
Mary Anne 29/10/1851
But also on the BMD I the same quarter as Mary is a male Halsted buried, no Christian name, can't find a birth reg or baptism for him, is it possible that Emily was a twin and the boy died at birth or was stillborn and he was registered at burial but nowhere else.
There may be a connection to the Tillbrook family as the 1851 census shows a Jos Tillbrook aged 70 born 1781 in Salcott Essex as a lodger with the Wicks family.
Just a thought Susanna may have kept Emily as she would have been a newborn and Susanna a daughter Louisa the same age by the 1861 census so Emily may have been a nurse child
Just had another thought, there was a thing called the Children's Friends Society in the 19th century, it was to send orphans and poor children to the USA and Australia/New Zealand, I notice a lot of the family seems to end up in Aussie could this be how they got there.