"The black-eyed was so softened by this deferential behaviour, that she caught up little Miss Toodle who was running past, and took her to Banbury Cross immediately."

My guess is that this is a reference to the nursery rhyme:
"Ride a cock-horse to Banbury Cross
To see a fine lady upon a fine horse.
Rings on her fingers and bells on her toes,
She shall have music wherever she goes."

Meaning that she crossed her legs, seated the child on the lower end of the upper leg, facing each other, holding her hands, and bouncing her as though she were riding a horse,
while singing the nursery rhyme to her.