Not only a delay to their response, the EMS has the full "tools of the trade" on board for the medical situation.
Flashing green lights have really no meaning so to speak, no one has to move over (most out of courtesy do), nor can the driver exceed the speed limit and can not go through a red light etc.. however EMS can do both of those things to get to a situation faster
Above is merely a suggestion/thought and in no way constitutes legal advice or views of my employer. www.OHTA.ca
On a more practical level, fire gets dispatched to medical emergencies to ensure a swift response, because EMS may be delayed for multiple reasons. Obviously, EMS was quick enough to catch up with you. In addition, an ambulance is probably quicker on any rural road than a pumper is, so why not let EMS get to scene first - pull the pumper over - and then see, upon arrival, if you can assist or clear. Something for the officer in charge of the pumper to think about. Hell, I pull over with the pumper or tanker when I see our own Chief flying up behind me in his Tahoe. Him being on scene before me is very helpful; why should I block his way?
On an even more practical level, friction between EMS and fire and police does occur; none of this is an excuse to act like total dic*s and have charges laid against each other. Most of these issues can be resolved through a civil conversation or, in your case, even involving the Chief to give you appropriate instructions.