Okay - after staying up all night last night, I've figured a few things out.
Ishmael Brierley is the illegitimate son of Alice Brierley.
Alice Brierley's father was Ralph Brierley - I deduced this from a) The 1861 Census, and b) the fact that Major Pickup and his wife were present at the second marriage of Ralph Brierley:
Marriage: 19 Feb 1865 St Clement, Spotland, Lancs.
Ralph Brierley - 34, Black Smith, Widower, Water Street
Susan Griffiths - 34, Spinster, Brown Hill
Groom's Father: Thomas Brierley, Slubber
Bride's Father: Charles Griffiths, Dyer
Witness: Major Pickup; Ann Matthew
Married by Banns by: W. N. Molesworth
Register: Marriages 1861 - 1873, Page 59, Entry 117
Source: LDS Family Search
As Ralph was listed as widowed in the 1861 census (with a 9 month old child) his wife would've had to have died around 1860-1861. I found a marriage record of a Ralph Brierley to a Matilda Ward in 1850, and a death record of a Matilda Brierley in 1861.
I'm still not sure how the Pickups fit into the whole thing, nor am I sure of other children from the Brierley-Ward marriage - since the '61 Census lists:
Alice 7 b Royton
Thomas 3 b Royton
John W 9 months b Saddleworth
but as of the 1871 Census, while Alice is listed as being 17, Thomas is now somehow 18.
Ralph Brierley Self M 40 Saddleworth, Lancashire
Susan Brierley Wife F 40 Manchester, Lancashire
Alice Brierley Daughter F 17 Rochdale, Lancashire
Thomas Will Brierley Son M 18 Rochdale, Lancashire
Emma Brierley Daughter F 5 Rochdale, Lancashire
Eliza Brierley Daughter F 3 Rochdale, Lancashire
Arthur Brierley Son M 3 Rochdale, Lancashire
Francis Mary Brierley Daughter F 1 Rochdale, Lancashire
If anyone can figure out how the Pickups fit in now that we have a bit more info to go on, it'd be muchly appreciated!