Does anyone have a theory on how Clara went from being a nanny to a teacher so fast?

Perhaps she'd trained to be a teacher before her character first appeared in the series, and she was just babysitting those 2 kids as a temporary job since she couldn't get a placement at the time. Or maybe she was babysitting to get money while training.

If you remember in The Name of the Doctor, Clara follows that Simeon guy into the Doctor's time tunnel in his grave. So basically, wherever the Doctor was in trouble because of Simeon, she was there to save him. She was spread out all over his timeline. She was born, she lived, she saved the Doctor, and she died. Simeon turned all of the Doctor's victories into failures, and then she jumped in and changed them back.