Praise be to Allah.
The shar‘i ruling concerning your question depends; there are three possible scenarios.
If doctors and specialist obstetricians say that it is most likely that your grandmother had nothing to do with the death of the infant during the birth, and that what she tells of the details that she remembers about the birth indicate that the death occurred naturally as a result of the difficult labour. In this case your grandmother is not liable and there is no sin on her; she does not have to pay diyah (blood money) or offer expiation because she is not responsible and had no connection to the death.
If the consultant doctors are uncertain and the details that the questioner related cannot help them to define the cause of death, then are your grandmother does not have to do anything, because the basic principle is that the individual is innocent and that cannot be changed without proof and evidence.
But if the doctors think it most likely that the cause of the infant’s death was a mistake during the birth, and that pulling on the head in this manner led to suffocation, then this is accidental killing, and the one who did that has to repent sincerely and fast two consecutive months as expiation for accidental killing. She also has to pay diyah (blood money) to the heirs of the infant unless they agree to let her off. The Prophet (blessings and peace of Allah be upon him) said: “The one who gives medical treatment when he is not known to have knowledge of medicine is liable.” Narrated by Abu Dawood and classed as hasan by al-Albaani. Saheeh al-Jaami‘, 6153.
Imam ash-Shaafa‘i (may Allah have mercy on him) said:
If someone orders a man to treat him with cupping, circumcise his child or treat his animal, and they die as a result of his actions, if he did what is usually done in the interests of the one to whom it was done according to those who have knowledge of such matters, then he is not liable; but if he did what is not usually done by one who is seeking to do the right thing and has knowledge of such matters, then he is liable. End quote from al-Umm, 6/185
Imam al-Khattaabi (may Allah have mercy on him) said:
I do not know of any dissenting opinion with regard to medical treatment; if (the doctor) mishandles the case and the patient dies, he is liable. The one who deals with matters of knowledge or practical matters when he has no knowledge thereof is a wrongdoer; if damage results from his actions, he is liable and must pay the diyah, but the hadd punishment is not to be carried out on him.
End quote from Ma‘aalim as-Sunan, 4/39
For more information please see al-Mawsoo‘ah al-Fiqhiyyah al-Kuwaitiyyah, 12/138; Ahkaam al-Jarraahah at-Tibbiyyah by ash-Shanqeeti, p. 519-534
To sum up, it is essential to consult specialist doctors. If they think it most likely that your grandmother made a mistake, then she is liable and must pay the diyah and observe the expiatory fast; otherwise she does not have to do anything. May Allah forgive us and her.
For more information, please see fatwa no. 114047
And Allah knows best.