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You have a serious situation here. Your sister may have an issue surrounding falsification of a birth certificate. She should check with a local attorney to see how the birth certificate can be rectified. It is very odd to see an incorrect birth mother on the birth certificate. I don not believe the courts will be too happy to see that mother and aunt conspire t falsify a legal public record,the birth certificate..
Under the law, a child born to a marred couple is presumed to be the issue of the husband unless the biological parent of the child contests paternity normally within 2 years of birth.. If your husband was not the biological parent then he may not have any claim to as the legal parent to the child.
There is no form that can be completed. If you legally want the child you will need to adopt the child. Normally an adoption can only occur if both parents agree, Your sister's husband/father does not have to agree and can petition the courts to take the child away from you..
Given the fraudulent birth certificate you have a substantial issue. There could be allegations of insurance fraud if you claimed the child on health insurance. You should contact local counsel to get the situation rectified.
There is no issue with you taking care or raising the child as long as it was drone with mother's approval and the child was properly cared for. By getting the Birth Certificate fixed and proceeding to adopt the child you may be able to avoid any further legal fallout.
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