Begum, a solicitor, has said she wants to "exercise her rights as a parent."
Muhhamed Raqeeb said he and Begum were thrilled by the High Court's judgement.
According to The Guardian, Begum said that "We have always had Tafida's best interests at heart and we have never wanted to come to court to have to argue for our rights to seek continued care in a world-class hospital [the Gaslini] willing to give her the treatment she needs. The entire experience of having to fight for our daughter's life over the last three months has been exhausting and traumatic for all of her family and we are glad it is now finally over."
She added that Tafida "is not dying and we are continuously seeing small but important signs that she is gradually improving and we have always been hopeful that she might make something of a recovery if she is just given the time and the right treatment to continue to improve."
Begum commented, "If it happens to me, I want my life to continue until a time that God actually takes me, not withdraw life support from me."
David Lock, a barrister for Raqeeb and Begum, said the judgement was an "enormous relief" and that the family "wanted to get on with the transfer" to Gaslini Children's Hospital.
And Paul Conrathe, a solicitor for the family, said MacDonald's ruling recognized "that a child's best interests are not merely medical, but include broader social and religious values. It also recognised the legal right of parents to request life-prolonging treatment in another EU state so that their child can be treated under that system of care and ethics."
He added, according to The Telegraph, that Raqeeb's parents "look forward to her receiving outstanding care at the Gaslini hospital in Genoa. They will also feel at peace knowing that Tafida will be cared for under the Italian ethical and legal system."
Bishop John Sherrington, an auxiliary bishop of Westminster and the English and Welsh bishops' representative on life issues, welcomed the ruling, which he said "has taken account of the wishes of Tafida's parents and their innate urge to do all they can to help their daughter in what are truly tragic circumstances. The heart-breaking illness of Tafida Raqeeb and the distress which the illness of a child causes parents touches the hearts of many people."
"I trust that all the medical professionals will cooperate to continue to give her the best possible care and appropriate treatment. Such international cooperation is essential good practice in the care of tragically difficult lives," he continued.
In July, Bishop Sherrington had urged that the public be reserved in judgement on Raqeeb's case, saying that "difficult dilemmas have to be faced. In that process, I hope that all due weight will be given to the wishes of her parents, while also respecting the clinical judgement of the doctors caring for her. Those of us not in possession of all the relevant information might best be reserved in our judgement."
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Katie Gollop, a barrister representing Barts Health NHS Trust, said MacDonald's ruling could affect other children.
Raqeeb's case follows similar campaigns by parents with children in NHS care. Charlie Gard, Alfie Evans, and Isaiah Haastrup were all gravely ill children on life-support treatment. The parents of each of those children lost court battles in recent years against hospitals that wanted to end their treatment, and prohibit them from being taken abroad for treatment.
The NHS trust cited both Gard's and Evans' cases in its arguments against continuing treatment for Raqeeb and against allowing her to receive treatment elsewhere.