The son of one’s full sister is a dhu rahm relative (a relative on the mother’s side) and the scholars differed concerning their inheritance. The correct view is that they may inherit if there is no one who is entitled to a share of inheritance (fareedah) or relatives on the father’s side (‘usbah). 

Among those who were of the view that they may inherit were: ‘Ali, Ibn Mas’ood, Ibn ‘Abbaas according to the most well known report from him, Mu’aadh ibn Jabal, Abu’l-Darda’, and Abu ‘Ubaydah ibn al-Jarraah (may Allaah be pleased with them). This is the view of the Hanafis and of Imam Ahmad, and the later Maalikis and Shaafa’is. 

See: al-Mawsoo’ah al-Fiqhiyyah (3/54). 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: The scholars (may Allaah have mercy on them) differed with regard to the inheritance of dhu rahm relatives, but the most correct view is that they must inherit, because the Prophet (peace and blessings of Allaah be upon him) said: “The maternal aunt is of the same status as the mother.” And he also said: “The maternal uncle is the heir of the one who has no heir.” And Allaah says (interpretation of the meaning): “But kindred by blood are nearer to one another (regarding inheritance) in the decree ordained by Allaah” [al-Anfaal 8:75]. The view that they do not inherit is a weak view. How can we deprive the maternal uncle or the mother’s father of their relative’s wealth then put it in the bayt al-maal to be consumed by people who are not related at all? 

Islam would not decree such a thing. The definitely correct view is that dhu rahm relatives do inherit, but only if there is no heir who is entitled to a fareedah share and no relative on the father’s side. End quote from al-Sharh al-Mumti’ (5/73). 

With regard to how the dhu rahm relatives inherit, the one who is present takes the place of his immediate ascendant, and takes his share of the estate. In the case mentioned in the question, the sister’s son takes the place of the sister, so he takes her share, which is half. Then the rest of the estate goes to him too, so he gets the whole estate, because there is no other heir.  

It says in Kashshaaf al-Qinaa’ (4/456): If there is only one dhu rahm relative, he takes all the wealth, because he takes the place of his immediate ascendant.  

See: al-Mawsoo’ah al-Fiqhiyyah (3/54). 

Based on this, if this woman dies, and she does not have any heir apart from her sister’s son, then he will take the entire estate. 

She has the right to bequeath no more than one-third to the person who raised him, or to anyone else, because of the report narrated by al-Bukhaari (5659) and Muslim (1628) that Sa’d ibn Abi Waqqaas (may Allaah be pleased with him) said to the Prophet (peace and blessings of Allaah be upon him): O Prophet of Allaah, I am going to leave wealth behind, and I am only leaving behind one daughter. Can I bequeath two-thirds of my wealth and leave her one-third? He said: “No.” He said: “Can I bequeath half and leave her half? He said: “No.” He said: Can I bequeath one-third and leave her two-thirds? He said: “One third, and a third is a lot.” 

The Messenger (peace and blessings of Allaah be upon him) allowed him to bequeath one third of his wealth, and the scholars are agreed that it is not permissible to bequeath more than one third. 

See: al-Mughni (8/404). 

And Allaah knows best.