Is the husband obligated to pay Zakat on the wealth of his working wife who has a salary, and is he sinful if she does not pay Zakat on her wealth?

The husband is not obligated to pay zakat on his wife's wealth. Rather, she is obligated to do so herself if the conditions for zakat's obligation are met. If the husband pays her zakat without her permission, it is not considered sufficient. The husband does not incur sin if his wife refrains from paying her zakat, but he must command her to perform what Allah has enjoined upon her, advise her, and explain to her the seriousness of withholding zakat.