I did address this at some point but it was a while back. There is no such thing as an emergency adoption. Emergency foster placements yes but not an adoption. Adoptions take time, it is a legal process. There has to be a home study completed and approved and there is a waiting period where the child lives with the family (typically atleast 6 months) to make sure everything goes well prior to an adoption being finalized.
However I only have experience with adoptions though foster care which I don't believe Natalia's was. I think she was privately adopted, possibly though a private adoption agency that does "second chance" adoptions, one of these has a Facebook page that posts pictures and descriptions of children needing another adoptive home,definitely worth a look for anyone interested in this subject. A private adoption would still have to have homestudy done and waiting period as far as I know and I would imagine a second adoption would be more closely scrutinized to make sure it was a better fit than the first adoption.
Since Kristine has stated it was an emergency adoption I don't want to say she lied but maybe didn't understand or exaggerated and what really happened is the Barnett's took Natalia as an emergency placement and the adoption was finalized at a later time when proper paperwork, evaluations and waiting period was met.