Finding Jane Hatton

[suttontrust]

I have the birth certificate of Ellen Hatton in Birmingham in May 1847.  Mother is Jane Hatton.  No father is given.  Okay, Ellen was illegitimate.  Jane and Ellen turn up on the 1851 census as lodgers in another household.  Jane is supposedly a widow, born c.1811 in Birmingham.  There is also Elizabeth Hatton aged 10, presumably another child of Jane.  I can't find her birth registration.
I can't find Jane on the 1841 (the Freecen version), either as a single woman or as a wife.  There is a death in 1857 which would fit, since Ellen appears to be a domestic servant in 1861.  The only Warwickshire Jane Hatton b.1811 on the IGI (apart from a member submission) wasn't born in Birmingham.
If a widow had an illegitimate child, would her status necessarily show up on that child's birth certificate?  Any ideas about finding Jane?