Wa `alaykum as-Salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother, let us first thank you for your question that shows your commitment to know more about Islam and its rulings and your sincere desire to put the Zakah money in the right place.
Indeed, the first people a rich Muslim should target with his Zakah and sadaqah (charity) are his relatives, especially those who are elated to his father or mother. But it the responsibility of every Muslim who pays Zakah to make sure that the recipients of this Zakah are really deserving it.
As for the case you are asking about, you should find a gentle way to make sure that your mother’s relative deserves the Zakah.
In his response to your question, Prof. Dr. Monzer Kahf, Professor of Islamic Finance and Economics at Qatar Faculty of Islamic Studies, states,
Of course, the worry is justified, you should be sure that your Zakah is given to a deserving person. The fact that the cousin is not married and has no son or brother is not sufficient. But also the fact that nieces and nephews moved in with her does not mean they spend on her.
You really should verify whether she deserves Zakah or not, maybe she collects rent from them which may supply her with sufficient income to remove her out of poverty. It is also possible that they do not care about her and they forced themselves on her, as a powerless woman, so she is still in need.
Allah Almighty knows best.