I think I found the correct family on the 1851 English census in Tyldesley civil parish, but the children seem to be a little older and no sign of a Patrick... maybe Patrick was born after this ?
The household is at Bridge St., and has Michael (40 - c1811) and Margaret (30 - c1821) and three children Henry (14 - c1837), Mary (12 - c1839) and John (8 mon), plus a lodger.
All the family, including young John, are listed as born just 'Ireland'. Maybe the census has his place of birth incorrect - did you locate an English birth cert for him, or another John ?
If that confirms Margaret's maiden as Fagan, then that should then that should help confirm the link back to that couple I mentioned in Meath. I'd be a little cautious of details from Ancestry for Ireland, as results for general searches can include details from submitted trees - which are often not accurate.