OP, I once had a female faculty member present with pretty severe chest pain. Given her age and history, my brain went to cardiac immediately. But I was still a new school nurse, and she was SO insistent I did not call 911 as she would send them away. She wanted to instead drive herself home. I would NOT let that happen and got her husband on phone, expressed my concerns to him and he was not able to convince her to let me call EMS, but did convince her to let him get her. Luckily, he was close by and took her straight to the ED, where it was later determined she was having a mild (thankfully) heart attack.
I should have 100% called EMS immediately and I would today, but I was green and not used to standing up to staff as much as I am today. (Not an excuse, of course, but what I saw later when I could re-examine the situation with a more experience under my belt.)
Staff stuff is harder here - they are adults and can say no. Let them send EMS away - but as I learned this year at work that my school will let me have the authority to not have a staff member return to work if I perceive their health issue will endanger students/staff and request they submit a doctor's note to me or HR. (Do I want this authority? That is a different thread, LOL...)