Does the questioner bear sin and wrongdoing for the death of her niece, who passed away following an operation in a public hospital, after the questioner had intended to pay for her operation in a private hospital but procrastinated in doing so?

It is obligatory to help those in need to save them from perishing, and whoever fails to do so while being able is sinful. However, the woman mentioned in the question is not sinful because the general hospital is presumed to be equipped, and the failure of the operation might have been due to another reason other than lack of equipment. We advise her to stay away from obsessive thoughts, maintain her adhkar (remembrances of Allah), and be keen on spending in the way of Allah.