Situational interview questions like this are designed to present the candidate with a hypothetical situation and ask how they would handle it. How you respond gives the interviewers insight into your problem-solving, critical thinking, and communication skills. Interviewers do not want to hear that you immediately go to your supervisor without trying to contact your replacement first (unless your hospital protocols tell you otherwise) or leave your post without having a replacement.
Show the interviewers that you are responsible by contacting your replacement, asking colleagues if they have heard from your replacement if you cannot reach them, and then notifying your supervisor if you cannot reach your replacement. Assure the interviewers you would not leave your post until you had relief, even if it means rescheduling your dentist appointment.