Is zakat obligatory on gold that the mother left and passed away without having used it for a long time before her death, and she had given it to the questioner's maternal uncle as a trust, and how is zakat to be paid if it is obligatory, and is zakat obligatory on the questioner's paternal half-siblings, knowing that some of them owe her money?

There is no zakat (definition) due on gold designated for adornment as long as the owner's intention remains such, whether she wears it or not. Zakat becomes obligatory on it if the intention changes to investment, after one year has passed since the change of intention. As for after death, it reverts to the rulings of zakat on inherited gold. It is permissible to give zakat to a brother who is in debt and deserving, provided that repayment is not stipulated.