Ok so I'm here to bring a little light to this whole situation and explain the diffrence between Animagi and Transfiguration.
Animagi is one of the hardest arts of magic to preform, it takes years to learn how to become an Animagus, really advanced stuff that a kid couldn't manage just by mistake. Anyway when a person has finally became an Animagus this person will be able to transform into an animal anytime anywhere without a wand, as easily as the person can transform back. The Animagus takes a form which reminds of the original humans personality. When transformed into their animagi form the human still keeps it's own mind, thus being the same as when human.
Animal Transfiguration on the other hand is like any other kind of magic. To be effective a wand is needed. Now the major diffrence between animagi and transformation is that when transformed into a cat, you will also get a cat brain, and since cat's can't use wands, you will not be able to transform back on your own, while an Animagus no matter how long he/she stays in their animal shape will always have the same intelligence as he/she had as human and will always be able to turn back to human. That is why Krum only transforms himself halfway to a sharke, because otherwise he'd not be able to turn back on his own and he'd most likly have eaten Hermione instead of saving her.
I hope that is all clear.
Now the answer to your original question. I don't believe that a young witch or wizard who haven't had any training in magic would be able to by mistake become an Animagus, even if this was possible the child would still remain human on the inside and keeping original memories.
I think it is rather likly though that the kid by mistake (if in great danger) could transform him/her self into an animal, then not being able to transform back. The kid would then also get an animal brain after a while forgetting he/she used to be a human.
Good luck with your story.