How is my mother to pay zakat on her money that she has not paid zakat on since 1999, given that her father used to receive it previously and her salary increased gradually?

There is no zakat on salary unless it reaches the nisab and a year has passed over it. The nisab for money is calculated based on silver, which is 595 grams. So, whoever possesses the equivalent of this amount and a year passes over it, zakat becomes obligatory on it. The easier approach is for a person to look at what they have saved when a year has passed over the first wealth that reached nisab, and then pay zakat on all of their possessions.

Whoever has neglected to pay zakat for years is obligated to pay it. They should strive to estimate what they possessed at the passing of each year, and the amount due is one-quarter of one-tenth (2.5%). If it is difficult for them to remember how much they possessed at each turn of the year, then they should act cautiously to clear their liability, along with the obligation to repent for delaying the zakat.