If you have either Edward's or Bedelia's parents names, can you tell us, please?
It would help the search!
Meanwhile, the 1851 census has this in Leeds:
Reference HO107, 3219, 398, 3
10 Brook Street, Leeds
McNulty, John Head Widower M 53 Hawker of Small Ware b Ireland
McNulty, Hugh Son M 20 Flax Dresser b Ireland
McNulty, James Son M 16 Flax Dresser b Leeds
McNulty, Michal Son M 14 Flax Dresser b Leeds
McNulty, Marla Daughter F 12 House Spinner b Leeds
McNulty, Margaret Daughter F 9 Scholar b Leeds
McNulty, Alice Daughter D 2 b Leeds
McNulty, Edward Son Widower M 29 House Dresser b Ireland
McNulty, James Grandson M 5 b Leeds
Does this family fit with what you know?