GRO Birth Registration
Henry Hadley, March Qtr 1839, reg. district Stourbridge Union, vol 18 page 481 - mother's maiden name Orford
Note - your describing "Lucy Maria, 1856, John William 1857 et al" as "Henry's siblings" did initially confuse.
Can find a couple of Henry's children (Harry Edward, Agnes Louisa) with mother's maiden name Teague on the GRO - but not Lucy Maria, first born.
FindMyPast has the marriage (transcription only) of Jane Teague, age 17 and Henry Hadley, age 19, on 5 Nov 1855. at Worcester. The Henry Hadley registered Mar Qtr 1839, would have been about sixteen and a half in Nov 1855 (assuming birth was registered shortly after his birth).
Lucy is listed on the 1861 census with Henry and Jane.
Lucy M Hadley Age5
Estimated Birth Year 1856
Relation Daughter
Father's Name Henry Hadley
Mother's Name Jane Hadley
Gender Female
Where born Worcestershire, England
I also have a copy of her marriage bans to George Mathew's listing Henry as her dad
Lucy Maria Hadley
Birth abt 1856
Marriage
16 Oct 1881 Birmingham, St Peter, Warwickshire, England
Spouse
George Matthews
Father
Henry Hadley
Henry was a shoe/boot maker