カナダの幼児関連の学校に通えばナニーとして働くことは可能でしょうか？
またはナニープログラムの条件を知っている方がいましたら教えてください。

まずはＣＩＣ（移民局）のＨＰをきちんと読みましょう。そこからです。
http://www.cic.gc.ca/english/pub/caregiver/index.html

You must have six months of full-time training in a classroom setting or twelve months of full-time paid employment, including at least six months of continuous employment with one employer in a field or occupation related to the job you are seeking as a live-in caregiver. You may have gained your training or experience in areas such as early childhood education, geriatric care, pediatric nursing or first aid. You may also have completed your training as part of your formal education. This experience must have been obtained within the three years immediately prior to the day on which you submit an application for a work permit.