Does the obligation to pay zakat for past years lapse on money saved for children's education that has not been zakated, and is it permissible to start paying zakat from now on?

You must repent to Allah Almighty for your negligence in paying Zakat (definition), for withholding it is one of the gravest sins. If the saved money reaches the nisab (definition) (threshold) (approximately 595 grams of pure silver or 85 grams of pure gold), then it is obligatory to pay its Zakat every lunar year, as long as it does not fall below the nisab during the year. Zakat is not waived by the passage of time. If you are unaware of when the money reached the nisab and the amount of Zakat due, then act based on your best judgment (taharri) and pay what ensures your certainty or strong presumption of discharging your liability.