It does seem that even from the limited evidence available that Hannah's father appeared to have a few different jobs before finally settling down as a chemical labourer.
On the 1851 census he appears to be listed as Pearl Button (?) Maker and on the 1841 there is a William Smith of the right age living in Gosta Green who is listed as the same. Wife is Sarah but the age is a little way out to correspond with the Smith/Wilkinson marriage so not sure. There are 2 children but there is not a Smith/Wilkinson reg. for either.
I don't like to advise people to spend money but it may be wise to purchase a copy of Hannah's birth certificate for more confirmation. And also Sarah's death certificate though there are a few it might be, Jun 1849 reg. Aston being the most likely. Digital images are just £3.00.
Also on the 1891 census William is listed as having had a stroke.