@mef, I'm sorry to hear that the date has been pushed back. Did you let them know that he would be open to an earlier date if something earlier opens up? The waiting is sometimes one of the hardest parts. I'm sure you feel helpless. Other health centers/systems will be busy too, but you won't know for sure unless you try. Is he open to looking elsewhere?
For Mayo Clinic, the process begins with filling out the appointment request found here: https://www.mayoclinic.org/appointments. Because of the urgency, I would probably follow up the form with a phone call. The appointment staff may be able to give you an idea of what to expect as far as the timeline goes. Hugs to you.