I might have found something:
1851 at Wavertree, Lancs (HO107/2192 1045 pg 3 #11)
John Smith 32 Joiner b Childwall, Lanc
Mary Smith 34 b Childwall
Thomas Smith 7 Scholar b Childwall
Elizabeth Smith 5 Scholar b Childwall
Mary Smith 3 (next page ... Ancestry bombed)
Charles Smith 1

1861: The family is still at Wavertree but Elizabeth and Mary aren't there any more (RG9/2737 232 pg 16 #87):
All born at Wavertree
John Smith 43 Joiner
Mary Smith 45
Thomas Smith 17
Charles Smith 11
Jane Smith 9
Alice Smith 5
John Smith 3

1871 still at Wavertree, born at Wavertree
John Smith 52 Joiner
Mary Smith 54
Charles Smith 21 Joiner
Alice Smith 16
John Smith 14 Messenger
Thomas Smith 17 Orphan Nephew Baker's Apprentice.

I haven't checked 1881 or 1841.

I couldn't find an Elizabeth Smith born Wavertree or Childwall in either 1861 or 1871, which was disappointing. If she is the right person, she's got to be somewhere. I had no luck with Lytham either.