I went into FindmyPast, which has a pretty good searching mechanism, for 1861 census when Elizabeth would be 16-18ish. I tried looking for any John Smith, joiner who had an Elizabeth Smith living with them. There were 10 but in all cases where Elizabeth was a daughter her age was way wrong. I did the same thing, replacing "joiner" with "carpenter" and there were 32 hits. I looked at them all, but none were good candidates.

1851 isn't ready yet for FindmyPast, so I thought I'd try Ancestry looking for Elizabeth Smith born 1845+-2yrs with a father named John. There were 183, with 20 of them in Lancashire. I started to look at specific people but for some reason the back arrow won't take me back to the list after looking at an image. I don't know what's happened here, but this is a big problem. So, I can't continue until I figure this out.