For what it's worth, in my experience, if you are so injured in an accident that you can't tell me who you are, you are probably what we call a "trauma alert," and I'm going to be far too busy and moving far too fast to worry about your cell phone, or even your identity.
Most likely, I'm going to remove or cut off all your clothes to check you for injuries, and put them into a red bag, so there goes your cell phone. I also try not to take any of your personal valuables with me in the ambulance, so if I see a cell phone, I'm not picking it up.
In the typical auto-collision situation, your identity will be determined by the cops, who usually are pretty diligent about getting your wallet or purse for your ID, including asking us for it, for their own purposes. They are also pretty good at then passing on the info to us and/or to the hospital slightly later.
Now, if you are found wandering somewhere with an altered mental status but no apparent injury, there's a bit more time to do a bit of "medical detective" work while we are assessing you.
Don't worry that a lack of ID will delay emergency medical care; it won't. Nor will your medical history or allergies be very important to know for your initial emergency treatment most often, although it can be very useful to help figure out what's going on with you, and may be essential for more definitive treatment or in unusual circumstances. Hospital ERs and ambulances can provide emergency treatment for you without the consent of you next of kin.
But all the above may be different where you are.
One thing that is likely to get noticed is a medic-alert or similar bracelet or necklace. If you have some special medical problems, then you probably should get one of these. The bracelet is a slightly surer bet, especially if worn on the left wrist. Why? Because most American ambulances have the most working room on the left side of the cot, so the paramedic will likely be to your left, start an IV in your left arm, etc. and it is more likely to get noticed.
It is also a good idea to carry a clearly labeled medical information card either directly behind your driver's license or in a highly visible slot next to it. Why? Because the first persons likely to go through your wallet are interested in your ID, meaning usually your driver's license, and will stop looking once they find it.
Nonetheless, I think having "ICE" on your cell phone is still a good idea, and checking for it is likely to become more common.
If you have medical problems, it is also helpful to prepare a paper with the following information on it, with copies kept at home, in your car, and folded up in your purse or wallet with the following information on it:
Your full legal name, address, and ALL telephone numbers;
Your date of birth, social security number and complete Insurance/Medicare/Medicaid information;
Full name, address and ALL telephone numbers for your TWO next of kin, and/or designated legal guardian or healthcare surrogate;
The full name, address, and telephone numbers of all physicians who are currently treating you and any specialists you wish to have treat you again if needed, and your pharmacy;
Any allergies you have;
A list of all medical conditions you currently have;
A list of all past hospitalizations, surgeries and major illnesses you have had;
A list of all medications you currently or occasionally use, including full name, dosage and instructions from the label;
A list of any holistic, organic, or OTC medications, nutritional supplements, vitamins or similar items that you use (just because no Rx is required doesn't mean it can't have major effects on your body or interact with other drugs or treatments);
A clear statement of whether you have an advance medical directive (a.k.a. "Living Will") and where it can be found.
It's best to keep this on your computer, too, for easy updating.
Jeff