Is Zakat obligatory for the years in which the father refused to pay Zakat, or only for the years in which the money became private property?

For zakat (definition) to be obligatory on wealth, it must be fully owned by a specific individual. The money your father deposited with you does not become subject to your zakat until you are informed that it is a gift to you. From that point, you begin a zakat year (hawl) for it and pay one-quarter of one-tenth (2.5%). As for the period the money remained with you as a trust before the gift, the zakat is obligatory on your father because he fully owns it. If your father has not paid it, you may pay it on his behalf, out of dutifulness to him and to absolve his responsibility, while informing him so he can make the intention for zakat.