Getting documentation of their medical condition and reason for not being in court is essential to trying to keep a warrant from issuing. It must be presented to the court. The public defender cannot help you unless appointed and is unlikely the judge will do so unless your family member is there to apply for services.
Please understand that answering this doesnâ€™t create an attorney/client relationship between us, and as hard as I try to answer your question well, it isnâ€™t legal advice. No matter how much information you put into a question, the answers you are going to get are still going to be vague. It is in your interest to contact a lawyer, most of whom will do a free consultation. Even 15 minutes with a lawyer is going to produce a more specific answer to your problem.
Immediately inform the Public Defender and bring paperwork in regards to the hospitalization and then the Public Defender can ask for a continuance with the Court.
Legal disclaimer: This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice nor does it constitute an attorney-client relationship. Thank You