Hi
I'm not sure - but I think you may have either the wrong baptism or the wrong census entry.
The 1861 census for William Buirley (Brierley) was for Derbyshire and both he and his wife are shown as born in Cheshire - not Lancashire. I don't think the district is Denton - I think it's something like Lenton or Linton
They have a son Joseph b c 1850 Cheshire (looks like Woodly). Looking at the 1851 census there is a Joseph Brierly aged 8 months b Bredbury Cheshire living with his grandmother Ellen Moors in Bredbury. On the 1871 census - their other son Robert b Glossop 1853/54 is living in Cheshire with Ellen Moors
Ellen must have re-married as there are children surname Bowler shown as her children.
From freebmd - marriage
William Brierley/Jane Bowler June qtr 1850 Stockport Cheshire Volume 19 Page 270
And also from Cheshire BMD -
www.cheshirebmd.org.ukCheshire Marriage indexes for the years: 1850
BOWLER Jane BRIERLEY William Stockport, St Mary Stockport ST16/7/138
However - on the 1871 census - William's birthplace changes to Gorton Lancs but Jane's remains as Bredbury Cheshire