I can't find a baptism in Birmingham for John Horrocks senior, but several in Lancashire. Worth investigating might be:
John born to a Walker Horrocks and wife Jane
John born to a James Horrocks and wife Betty nee Walker and
John born to a John Horrocks and wife Ann nee Walker
I have found some of the children of John and Hannah but not yet investigated any of them in depth.
Two of the published trees have John born to Walker Horrocks and his wife Jane, nee Ogden. In the hope that this is backed by some peersonal family history/documentation this is the couple that I will investigate first.
As I have said upthread, the difficulty is proving that any John Horrocks who later appears in Birmingham, is the same John Horrocks that originated in Lancashire.
You've been a great help. Many thanks for your time and trouble.