Just come upon this again - For information to those who kindly helped me in 2012: I got the certificate, "my" Ann wasn't the child of William Newbold and either Miss Basford or Miss Parkes.
She was a daughter of William Newbold an agricultural labourer, and his wife Harriet Topliss, but because for some reason Mrs Newbold is not on the 1861 census under Harriet Topliss, but as Harriet Brailsford, and young Ann appears there under "Nanny Brailsford" it took me a long while to track her down! I was able from the certificate to get a lot further back with both families, though, despite the continued proliferation of Williams Newbold.
It looks as if either she left her William, or he did die between the two censuses. There are other children born between the two censuses, 1851 and 1861, when she is with another man,so although I'm fairly sure Ann is in fact the child of William, I'm less certain about the parentage of the others.
The Joseph born two years before Ann could well be the brother who was a witness at Ann's marriage, and there is a younger brother to Ann, called William, so I suspect he may well be William's child too, but then the next born is a Samuel in 1858ish - and there are Samuels in both the Topliss and the Newbold families, so not a lot of help there.
I haven't managed to find a legit marriage for Harriet Newbold as a widow to the John Brailsford she hooks up with - they stay together for a couple of censuses whilst I suspect he's navvying on the railways, all over, haven't found a death for John Brailsford yet (and then she morphs into "Harriet Robinson" in Idle from 1881 on to her death, although here again I've not found a marriage to her John Robinson, in fact I think I've found him a wife and children over in Hull)
If she didn't just hop off with John B., leaving William, that leads me to think I need to find a death circa 1857+ for William Newbold. I did wonder about an 1861 census entry for a William Newbold ag lab, married, of not too far from the right age, lodging with Hannah Marple and her family in Derbyshire, but that's a bit clutching at straws.
Thanks again for the help and time and effort all you spent on this. It just cropped up when I was searching for something else, so I felt it was worth updating.