Is it permissible to pay Zakat to a project for establishing a hospital for treating children's cancer, and what is required if it turns out not to be permissible?

Building hospitals is not among the expenditures for Zakat. The amount your mother and sisters gave from their Zakat to the hospital does not absolve their duty, and they are required to pay Zakat again, unless they relied on a fatwa from a scholar regarding this matter. In that case, we hope their duty has been absolved.