Maybe the oxygen had a substantial positive effect, or maybe the child would've recovered on her own. We really don't know, since there are other reports of children who have good neurological outcome despite terrible prognosis [1] [2].
I'm suspicious because of the unusual and/or stereotyped responses in the Medical Gas article and the linked YouTube videos: "doctors said she had 48 hours to live" (doctors don't say things like that) and "this demonstrates that we're inducing 8101 genes!" (ummm, OK...), etc.
Also, be suspicious when something like this hits all the pseudo-news sites simultaneously. It reminds me of the articles that go something like "16 year-old cures cancer...DOCTORS HATE HIM!".
Finally, I'm very happy this little girl has been given a second chance and hope for her continued recovery. However, don't forget that a toddler was left unsupervised and submerged in a pool for 15 minutes. Some people call that an accident; some people call it neglect.