Is Zakat obligatory on a sum of 16,000 Saudi Riyals that I lent to my brother, and then permitted him to spend it on my mother after two years had passed?

The more correct view is that zakat on a debt is obligatory upon the creditor in all circumstances. In this case, the zakat on the money owed to you by your brother is obligatory upon you for the past two years. Your permission to your brother to dispose of the money is considered a constructive possession. However, if your brother is a procrastinator or a denier, then zakat becomes obligatory upon actual possession for the past years, and some scholars hold the view that it is only obligatory for one year.