Praise be to Allah.
If there is a full brother or a half brother through the father, then the siblings’ children do not inherit.
Based on this, then the children of the brothers, in this case, do not inherit anything.
If there is a full brother, then the half-siblings through the father do not inherit anything, whether they are male or female. There is scholarly consensus on all these points.
See al-Mughni, 9/22, 23
Based on this, the estate in this case should be divided among the full-brother and the full-sister, with the male getting twice the share of the female. The evidence for that is the verse in which Allaah says (interpretation of the meaning):
“They ask you for a legal verdict. Say: Allaah directs (thus) about Al‑Kalaalah (those who leave neither descendants nor ascendants as heirs). If it is a man that dies leaving a sister, but no child, she shall have half the inheritance. If (such a deceased was) a woman, who left no child, her brother takes her inheritance. If there are two sisters, they shall have two-thirds of the inheritance; if there are brothers and sisters, the male will have twice the share of the female. (Thus) does Allaah make clear to you (His Law) lest you go astray. And Allaah is the All-Knower of everything”
And Allaah knows best.