Before anything else, ask yourself two things:
Would this person possibly benefit from (A) an automatic external defibrillator (AED) or (B) Narcan[1]?
Can the person safely get into a car and be driven to the hospital?
If the answer to 1A or 1B is YES: Don't drive. Call 911, clearly state "cardiac arrest," or "overdose" give your exact location, and start effective CPR if required. A police officer with an AED will likely arrive quickly. Getting the location right is critical—this is life-saving information.
If 1A/1B don't apply and you can answer YES to question 2: You have some thinking to do. I suggest doing it now, in advance, whenever you move—think through how you'd handle massive bleeding, heart attack, or stroke symptoms.
It's worth briefly considering emergency scenarios and the risks you're willing to accept. Ambulances or fire engines sometimes can't reach you quickly: logistical issues, mechanical failures, dispatcher problems, insufficient volunteers. In rural settings during a cardiac event, waiting thirty minutes for basic EMS care—followed by a 30-60 minute hospital transport when you have alternative transportation—may not be your best choice. Even in a volunteer live-in program with career-grade response times, I found it could take 20-30 minutes to reach people at the edge of our territory. That's not counting the 2-3 minutes to get us awake and out the door at the station, plus another 2-3 for dispatch.
My household is minutes from two decent suburban EDs (we're lucky). Certain situations would lead us to skip 911 and drive straight to the ED: massive hemorrhage or an obvious heart attack when another adult is present to drive. This requires nuance. Time saved by skipping the ambulance can easily be lost to an incompetent admissions screener. You need to use the magic words: "heart attack," "chest pain," "think I am going to die." If you're having a stroke, you may not be able to drive at all (and you shouldn't). You'll also need to choose the right hospital—challenging in the moment, potentially impossible if you're impaired. The wrong hospital can be as lethal as waiting for an excessively delayed ambulance. In large cities with saturated EDs, this strategy often doesn't work: too many false alarms and just overall volume mean you won't skip the line.
I have direct experience managing and assessing these issues. You may not—consider getting meaningful first aid training. It helps.
The general rule: If you're confident in your department and know a nearby fire station generally has a paramedic-engine or paramedic-staffed ambulance with reasonable response times, wait for it - paramedics can do a lot for you on the way to the hospital and most critically get you to the best facility for care. You could crash your car or deteriorate en route to the point where you can't drive. But if you're fifteen to twenty minutes in and don't hear sirens (admittedly, not all departments use sirens properly), it's time to consider leaving—and how you're going to do it.
[1] Regarding Narcan: I won't engage in broader discussion about police possibly asking about circumstances requiring it—that's your business. IANAL. But many cops and almost all ambulances carry it, and the person will be alive after they administer it.