So this is your relative-
Marriage: 24 Nov 1862 St Peter, Liverpool, Lancashire, England
Alexander Baines - (X), full, Engineer, Bachelor, Bridgewater St.
Mary Ireland - full, Spinster, Bridgewater St.
Groom's Father: John Baines, Wheelwright
Bride's Father: Thomas Ireland, Engineer
Witness: Thomas McMasters; Elizabeth McMasters
Married by Banns by: W. H. Denny Curate
This might be a brother-
Marriage: 14 Nov 1860 St Chad, Claughton, Lancashire, England
James Baines - full, Farmer, Widower, Claughton
Alice Beaumont - full, Spinster, Claughton
Groom's Father: John Baines, Wheelwright
Bride's Father: James Beaumont, Wheelwright
Witness: Joshua Procter, (X); Rebecca Chaine; John Chaine
Married by Licence by: J. Chaine, Rector
You say this-
"I've found an Alexander Baines in 1861 Newton,Lancs. (RG9/2787 pg.37), a boarder, 20 yrs. old,an iron foundry laborer,born in Sheffield,Yks. No birth record has been found in Sheffield for this Alexander. Boarding along with Alexander is a Hugh Baines,46 yrs. old, a blacksmith, born Hulton,Lancs. I don't know if this is the Alexander I'm looking for or who this Hugh is.."
I'm currently wondering if this Hulton (and there are places called Hulton) is actually Halton which is about three miles down the Lune valley from Claughton.
Further to this - there is an older married couple a James and Mary Baines, both born at Claughton living at Haydock which is only up the road from newton le willows (formerly newton in makerfield).
Baptism: 31 Jul 1803 St Chad, Claughton, Lancashire, England
James Baynes - son of John Baynes & Nancy
Born: 9 Jul
There's a father and daughter John and Rebecca Baynes, both born at lancaster and living in islington, liverpool.
There's definitely some sort of flow of people called Baines from the Lune valley down to Merseyside.
Marriage: 11 Oct 1794 St Mary, Lancaster, Lancashire, England
John Baines - Carpenter, Widower, Lancaster
Nancy Beaumont - (X), Spinster, Caton
Witness: Wil'm Lucas; Eliz'th Lucas, (X)
Married by Licence with consent by: J. THOMAS, Curate
More to follow (hopefully) I'm just sticking my conjecture here so I don't lose track.