Yes, a great find, wilcoxon.
So now we know when John William died.
So if Margaret was illegitimate, then maybe the John Smith burial at Philips Park in 1872 is the right one. I would have thought that Mary would be buried there as well, probably in a common grave?
Speaking of Broughton!
Playing around with a godparent search again (despite myself!)
What is this all about? 
From Lancs OPC
Baptism at Bradford St Brigid, 7 Nov 1880
Joanna Burns, filia Jacobi Burns & Mariae (formerly Smith)
Born 24 Oct 1880
Abode 56 Gibbons Street
Godparents Arthur McCandrew?, Margareta Welsh
Notes: Ipsa matrimonium contraxit die 3 mensis Maii anno 1919 cum Leonardo Broughton in hac Ecclesia Stae Brigidae apud Bradford. Praesentibus testibus Jacobo Burns et Helenae Broughton. J. C. Holland
Not to harp on, but another link above which ties both families, and makes it seem unlikely that James Burns & Mary Smith nee Byrne/Burns were siblings.
Baptism above of Joanna or Annie/Anne as later called, married Ernest Hesford 1902 and then Leonard Broughton in 1919. Annie can found in census with parents in 1891 see reply # 60 by John. Also on census with the Holland family as Anne, sister to Catherine. Birthdates match on pension record of death of first Hesford husband, and marriages all link see reply #65.
Anne Broughton buried with John William Smith and sister Margaret Pendlebury nee Smith (Margaret baptised Burns, reply #51, by John) Burial for Anne and others link in reply #63 found by Wilcoxon.
Godparent in above baptism, Arthur McAndry/McAndrew was 3mths in 1861 (see reply #24 by Wilcoxon) by 1879 became stepson of Honora/Ann McAndry nee Byrne.
Godparents were normally chosen around the same age/birth decade as parents, or young enough, who could then step in if parents were absent or died. (and still be young) Not sure if it was one from each side or random.
Not looked for the Welsh connection or Smiths. Best on new thread for those, if intended, as lengthy threads can sometimes have info lost or is hard to find, or take in, amongst a lot of posts.
Just a few thoughts from this lenghthy thread research. FH can be costly, but if able at some point, I think the 1879 marriage for James Burns & Mary Smith could be informative.
Both marriages 1864 & 1879 fathers should hopefully match, and confirm. Also if Honora/Ann Byrne is a sister to Mary, it should be same, unless a curve ball drops.