Since Sidebottom existed as a surname in Tintwistle there is a possibility Mary married someone local to the village. A search of Sidebottom marriages in St. Michael Mottram-in-Longdendale on the IGI married to Marys gets
JOHN SIDEBOTTOM  
MARY WOOD  
Marriage:  24 OCT 1830   Mottram In Longdendale, Cheshire 
JAMES SIDEBOTTOM  
MARY FIRTH   
Marriage:  12 JUL 1824   Mottram In Longdendale, Cheshire 
 
SAMUEL SIDEBOTTOM  
MARY HOWARD  
Marriage:  12 JUN 1821   Mottram In Longdendale, Cheshire 
JOSHUA SIDEBOTHAM  
MARY WINTERBOTTOM  
Marriage:  18 APR 1808   Mottram In Longdendale, Cheshire 
If any of these is the correct marriage and it is a bit of a long shot, the earliest marriage would be the likeliest.
Joshua and Mary had these children baptised in Mottram.
SIDEBOTHAM, Charles   
Christening Date:   24 Mar 1811
Margaret Sidebottom
Christening: 10 APR 1814 
SIDEBOTTOM, James   
Christening Date:   10 Dec 1815
Jemima Sidebotom
Christening: 28 JUN 1818 
Betty Sidebottom 
Christening: 25 FEB 1821 
Kitty Sidebottom 
Christening: 03 AUG 1823 
Christiana Sidebottom
Christening: 01 APR 1827 
Joe Sidebottom
Christening: 05 JUL 1829 
Mary Sidebottom
Christening: 02 SEP 1832 
This appears to be the family in 1841
HO107 110/2 folio 17 page 28
Wagstaff Street Dukinfield  Cheshire (like Marple a parish in Stockport)   
Mary Sidebottom 55  Cheshire
Ann Sidebottom 33  Cheshire Stretcher?
James Sidebottom 25 Not born Cheshire Corder?    
Jemima Sidebottom 20 Not born Cheshire
Betty Sidebottom 20  Not born Cheshire 
Betty Sidebottom 18 Not born Cheshire 
Joe Sidebottom 12  Cheshire
Mary Sidebottom 9 Cheshire 
The unmarried Sidebottom daughters are together in Dukinfield on the 1851 census (it could just mean their mother had died).
1851 census HO107 2237 folio 341
Wagstaff Street Dukinfield  Lancashire   
Ann Sidebottom  42  Mathern, Cheshire,  Head  Cotton Piercer?
Betty Sidebottom  30  Stalybridge, Lancashire, Sister  Cotton Piercer
Kitty Sidebottom  28 Stalybridge, Lancashire,  Sister Cotton Frame  ?
Mary Sidebottom  18  Stalybridge, Cheshire, Sister Power Loom Weaver 
All very interesting but absolutely no evidence of a Hannah Winterbottom marriage to a William Brierley and also tricky to find both a Mary Winterbottom and a Hannah Winterbottom of roughly the right ages baptised in Mottram with the same parents, where one of them doesn't die young, so this Sidebottom family remain an unproven possibility.
Not sure what look up you are requesting for a search on Ancestry. If you search on Sidebottom you are searching on the woman's maiden name and Sidebottom was Mary's married name.
Regards
Valda