The Jan 1838 St Peter's, Bolton baptism shows James Haney Mather was the son of William a weaver and Jane. There only appears to be one William and Jane (Wilkinson) marriage and this fits with William being born in Bolton following his siblings born in Ainsworth and Droylsden. No other sibling was baptised in Bolton.
In 1841, James Henry is missing, not with mother Jane at Drolysden, Manchester (so may not belong here, although father William is also missing).
In 1851, James (17) was an Apprentice clogger in lodgings followed by a clogger in 1861. In 1861, another James Mather (aged 19) was a clogger, son of Isaac Mather at Wittaker's (i.e. Whittakers, Entwistle). That James died in 1861.
In 1861, James H Mather (27) was a Clogger at 49 Matthew Street South, Bolton. His wife Martha (32) was born Knutsford.
About 1863 son Henry a coffin maker was born, possibly in Ireland. Henry married in 1883, naming his father James H Mather a shopkeeper.
From 1865 to 1871, James and Martha may have been in Ireland (as daughter Ellen was born there about 1865).
In 1881, James Henry Mather (47) was a Provision Dealer, living at 66 Moor Lane, Bolton. With him were wife Ann (47), John William Mather (21) and Ellen (15).
In 1905, James H was an Undertaker at the time of his death (and his son Henry a coffin maker).