Seems like an absurd policy. Nothing happens in under 15 minutes in a hospital. What kind of delay is the patient causing? They need to give you a 30 minute heads up or an estimated time if they plan to require a 15 minute turn around. Most procedures in my hospital (which is large) take more than 15 minutes to even GET to never mind get ready, in the chair or stretcher, and taken down. If you know a patient has to go at some point, all paperwork and prep work should be done. We also do not have a transporter, so most of the time patients are taken down by a tech or aide, so if the tech or aid is with another patient, that causes a delay.
I think your policy is unrealistic. But I guess it depends on your facility size, do you have a transporter, what is causing the delay. You need to tell the patient they need to go now. Only delay I can see on their part would be if they were using the restroom or need pain medication. You will need to manage the patients expectations too. In the morning tell them, you have this procedure and when they call you will need to leave immediately. So they know.