Hi
The most interesting thing is that the father on Emma's birth certificate is called George! Which is the name that Elizabeth gave for her father when she married.
So it does somewhat call into question the info Jane gave when she was in the workhouse in 1860/1. Then she said she had been married to Edward Edwards. Unless perhaps he was George Edward Edwards!
What to make of it? How much of the truth was she telling? The facts that she gave about herself seem to be accurate, other than being a Smythe (or Smithe even) rather than plan old Smith.
No birth has been found for Elizabeth, so there seems to be a bit of an impasse. Yes, Fanny Elize was my mistake (sorry!), it was just a coincdence (as Mabel said)
You could perhaps look for a death for Jane, possibly before 1881, possibly in St Pancras. And Elizabeth (and Louisa) should be around somewhere in the 1881 census. As we can't really be sure of her age, and she may be a servant, possibly just born London, you may not be able to pin Eizabeth down with certainty.
There is an Elizabeth F Edward, 18, born St Pancras, a servant in St Giles. Could that be some kind of name reversal of the Fanny E from 1871?
Did Elizabeth give herself the Alice name later? Hard to know.
Jon