Dfyjkt

France has at least three numbers which can be called in case of a medical emergency:

• 
  15 (SAMU), handled by hospitals


• 
  18 handled by firemen


• 
  112 handled by someone else, or maybe one of the above

There is also 3624 (SOS Médecins) for medical care (not emergencies)

Which of these numbers should be called to minimize the handovers between the services and maximize the chance to get the service which will actually provide the help? (I would say 15)

Which of these systems are more likely to have English-speaking operators? (I would say 112)

The first thing to note is that all services communicate with one another. So if you call the SAMU saying you need help with a fire then they'll reroute you to the fire department. Now having said this the answer is: it depends on the nature of the emergency. Obviously call the fire department (18) if there's a fire and the police (17) if there's a crime. Regarding health related emergencies the rule of thumb is to call the SAMU when it's a medical emergency, and call the fire department if there's a medical/life-threatening emergency requiring special equipment. For example, you're in a bistrot and someone feels unwell: SAMU. You're walking down the road and you see someone trapped under a heavy object which can't be displaced: fire department (oh and also don't move the object of you might cause septic shock). The reason is simple: SAMU brings ambulances, paramedics and maybe doctors; fire department brings ambulances, paramedics, maybe doctors, and specialised tools.

Now, in some cities firemen are actually running the SAMU. This is the case in Paris and maybe Marseille if I'm not mistaken. The habit of calling 18 rather than 15 for medical emergencies might come from this. It might also be due to the fact that in general the emergencies one might witness on a daily basis often require the specialised action of firemen: fires, vehicle accidents, work-related injuries, drownings, etc.

For English-speaking support 112 is the way to go. Note that 112 interfaces with the local emergency numbers anyways. Also, since English-speaking emergencies add an additional level of complexity to the matter, it's safe to assume that services might take longer to be activated if one goes 112 rather than 15/17/18.

For reference here's a handy list of French emergency numbers and explanations (in French) and even the governmental page on emergency numbers (also in French).

112 is your best bet for life threatening emergency services across the EU.
It's the equivalent of the US 911 system, with regional centers that will dispatch the closest applicable first responders to your location.

Fire services in the EU don't generally handle medical emergencies (unless they develop in the line of their regular business or they're on scene when the emergency develops).
Same with police.

Calling a hospital would most likely lead to the hospital putting you on hold for triage by the phone hotline of their emergency room, who then will decide whether you should come over, call a non-emergency doctor, or are in need of an ambulance.

In general, that's what you'd do in most of Europe when you are in need of quick medical attention outside of office hours of family doctors/general practitioners and it's not urgent enough to have ambulances and/or other emergency crews rush to the scene with all lights and sirens blazing because every second counts towards saving a life in imminent danger.

Do remember that abuse of the 112 system can be a crime.

The 112 is probably your best bet. It's available across the EU and handles all kinds of emergencies and dispatch the appropriate response teams. French people are more and more encouraged to call this number since it minimizes the risk of "calling the wrong service".

The 18 is also a good bet. As you know, they call the fire-fighters but they handle injured persons as well. For example, if there is a car crash, fire-fighters will dispatch an ambulance of theirs to take care of the injured and bring them to the hospital if necessary.

The 15 is for severe injuries. The SAMU can be dispatched in car crashes for example, but mostly when there are severe injuries.

Since these last 2 numbers are national, it is less likely that you will speak to a person who is skilled in English (prepare for ze Frenche accente).