What is the status of an illegal mother who has a disabled American baby?

The short answer is none, unfortunately. If she is from an underdeveloped country or a country where medical treatment for the baby would be difficult or unavailable, and she were placed into removal (deportation) proceedings, she could apply for a waiver that her removal would subject the baby, a U.S. citizen to extreme hardship. She might also, in that situation, be eligible for cancellation of removal if she has been in the U.S. for a long time. She should seek legal advice from a non-profit like Catholic Charities. Depending on her individual facts, she might have some other claim to legal status. Her Immigration status should not prevent her from seeking government benefits for the U.S. citizen child, such as Medicaid. Under the Americans with Disabilities Act, public schools also offer some really great services to disabled infants and toddlers that she should take advantage of.