What priority if any is given to Emergency Services in flight clearance?
I work as a fixed-wing pilot for a hospital-based Air Ambulance program. Let me add my experience to the ATC orders accurately cited by Pondlife's answer: the MEDEVAC call sign (the old "Lifeguard" is still used when filing the flight plan) indicates a need for priority handling.

ATC is often helpful in providing us with priority routing. This includes giving us direct clearances, moving us up in the landing que, allowing us an unrestricted approach speed, and otherwise expediting our flight. However, ATC does not always provide priority handling, and I have often experienced ground delays of 5-10+ minutes as ATC apparently prioritizes other traffic, including air cargo and the like. This is rarely a big problem as patients are typically stable, but it can be frustrating when priority handling is expected.

However, if medical staff determine that a patient is in critical condition, we will announce this to ATC—typically accompanied by a request for a specific clearance or change to a previous clearance. This announcement re-iterates our need for priority handling and indicates the urgency of the situation. In this way we can work with ATC to best care for the patient. We will often make such an announcement when checking on with approach control for their information with no specific accompanying request. Occasionally, however, we will need to request such priority handling because ATC has issued us speed restrictions or delaying vectors to put us behind other traffic. ATC is very good at working with us when we do need the priority handling.

In other words, ATC usually automatically gives us priority handling, but when they don't—and we need it—we ask for it and they are very good at working with us to provide the priority handling.