If there is only one relative, he takes the whole estate

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.