She does not have a conviction, bednobs. It's a caution.
m1, I can't find much about expunging police cautions, other than that it's next to impossible. From the IPCC web site:
"If you would like to challenge a police caution
or fixed penalty, or contest an ASBO you should contact
your local Citizens Advice Bureau or seek legal advice"
and
"Any disputes regarding the
information on your Criminal Record Disclosure should
be raised directly with the Criminal Records Bureau."
It seems a pretty hopeless situation, sadly. If you think you can provide absolute proof that you did not do what you tacitly confessed to in accepting your caution, then you might conceivably be able to have it overturned, I should imagine, but other than the word of your daughter, who presumably could be coerced into giving evidence via threats of putting her up for adoption, I can't really imagine any way of proving it.