It would be uncanny if there was also an Anderson connection, but it's possible, although I think unlikely.
My Lawrence and Hannah Hartshorne had a daughter Phoebe Hartshorne Baptised 31 December 1837 in Tipton. She married a John George Manders(b. 1838) who died about 1866. She then married John Anderson(b. 5 Feb 1839 Coxhoe, Co. Durham) in 1867 at the Durham Register Office. John's family were all from Co. Durham and Phoebe settled in this area of England.
I do not have a Luke Anderson in my tree, and it is the wrong end of England, but it is possible. Luke could have been a cousin. Do you know any more about Luke's background?
With regard to the Hartshornes and Tipton, do you have any further information? I feel sure that your George and my Lawrence will have been brothers.
Your Eliza Hartshorne(b. 1838) is obviously different to my Eliza Hartshorne(b. 1831). I think they were first cousins.