There's another possible 1861 entry in the Newcastle workhouse: RG9/3817 fo80 p22 listed next to a Hannah Graham aged 9. Since the other one was in Barnard Castle, this one might possibly be more likely.
However, although ultimately it would be important to find census entries so as to get the complete picture and to corroborate as much as possible, there are other things that can be done.
If, for example, Margaret's father died when she was young, she may not have known exactly what his name was, hence "John William" on her marriage certificate when he might have been just William. Being orphaned could also explain her being in a workhouse in 1861, if that is her. Who were the witnesses at Margaret's marriage? They might provide useful clues.
Also, from Margaret's birth certificate, what was her mother's full and maiden name? That should help in tracking down Margaret's parents' marriage, which might then help in pinning down exactly who her father was and when he was born. If the Elizabeth who is wife in the 1851 census was William's first/only wife, it looks as though the marriage might have been around 1825-1830; this means there won't be a certificate, only a parish register entry, but that's better than nothing. And at least with children born after 1837 we can be reasonably sure of finding the right marriage because of having Margaret's mother's maiden name.
Arthur