Will I be sinful if I do not pay the zakat on my money while I am a minor, and I relied on my guardian brother who delays paying it? And will the delayed zakat be waived for me after I am no longer under guardianship, or will I be responsible for it?

The one responsible for the zakat (definition) on a minor's wealth is the one who possesses the wealth (the guardian or trustee). He must disburse it if a year has passed and the wealth has reached the nisab (definition) (minimum threshold for zakat). If the guardian or trustee does not pay the zakat, then the minor, upon reaching adulthood, must pay the zakat for the past years for which it was not disbursed; for the statute of limitations does not waive zakat. You may pay it from your own personal expenses after knowing the amount of wealth and what is due on it, and coordination with the trustee is necessary to avoid paying it redundantly.