One solution would be to split up the paramedics. There are always two to an ambulance now, one to drive and one to take care of the patient in the back on the return trip. If we modified a motorcycle and put one of these paramedics on it he'd be able to speed through the narrow gaps in any intersection that was blocked and reach the victim much faster. It wouldn't be long before people learned that this motorcycle would quickly be followed by the regular ambulance, giving them time extra time to move over.

At the accident scene (or house if that's where they were called to), the driver would help the motorcycle patient load the patient into the ambulance and then put the bike on a lift attached to the front of the vehicle before heading to the hospital. We might have to add some kind of counterweight to the back of the ambulance to keep it from becoming top-heavy on the front-end, but this is doable.

Also, the motorcycle paramedic should also have a radio built into his helmet in case he needs to get, or give, updates along the way.