What is the legitimate distribution of inheritance for a man who left behind paternal cousins and sisters' children, and what is its legitimate evidence?

The heirs are the male paternal cousins, as they are the 'Asabah (agnatic heirs). As for the sons and daughters of the two sisters, they are from the Dhawu al-Arham (relatives by blood who are not entitled to a share of inheritance as a Dhu Fard or an 'Asabah), and they only inherit if there is no Dhu Fard (sharer) or 'Asabah. The proof for this is the saying of the Prophet, peace and blessings be upon him: "Give the prescribed shares to those who are entitled to them, and whatever remains is for the nearest male relative."

Inheritance matters are serious and complex, and a fatwa alone is not sufficient. Rather, they should be referred to Sharia courts for investigation, as there might be an heir who is not known except after investigation, or there might be wills, debts, and other rights that take precedence over the heirs' right to the money.