The 1861 Census record for 112 Mill Street established the connection to Elizabeth Cannell. She is given as John Purse's 72-yo mother-in-law (so presumably Lydia Purse's mother), and a widow. Also under the same roof are:
Lydia Taylor (17), apparently John's niece, and
Elizabeth Cannell (16), also apparently John's niece.
All residents are listed as being born in Liverpool, but I think this should be taken with a grain of salt.

At face value, 16-yo Elizabeth should be daughter of unknown male Cannell (Lydia Purse's brother), and also grandaughter of 72-yo Elizabeth Cannell.

As for Lydia Taylor. "Lydia" is a Cannell family name, which suggests an unknown female Cannell, sister of Lydia Purse, who married unknown male Taylor. (Note however that John Purse's father is thought to be a tailor.)

On the other hand, Lydia Purse was old enough to be the biological mother of one or both "nieces".