Types of Emergency
The most probable causes of an emergency are Fire aboard, leak aboard, or life threatening illness aboard. A failure of Life Support is also a dire emergency...
Fire and leak are the two most emergent of these.
In order to render the need for a rescue flight, you would need to damage a Soyuz, possibly two.
Self Rescue via Soyuz
At any point, the station is to have no more persons aboard than it has return capsules for.
This means, most of the time, there are two soyuz capsules docked.
The Emergency Ops Plan is pretty clear that plan 1 in an emergency is "Tell NASA", priority 2 is Stay alive, and priorty 3 is "Fix it if you can, come home if you can't fix it." Most of the outcomes on the flow chart on page 7-1 involve leaving the station in a Soyuz capsule.
Self-Rescue via cargo ship.
At times, there is also a cargo capsule docked - either the Dragon, a Cygnus, or Progress capsule. In a pinch, any of these three could be used as a lifeboat, as well... but more drastic, because it would require (1) travel in a space suit, and (2) inadequate cushioning of note for the astronauts. The cargo units are pressurized, but don't have the same atmospheric processing as manned capsules. They have several person-hours of breathable air, presuming, of course, that the air aboard isn't contaminated by the emergency.
SpaceX to the Rescue
At KSC, SpaceX has rented a hangar. At KSC, SpaceX is testing the Dragon Manned Capsule.
At any given point, SpaceX seems to have a Falcon rocket nearly ready to launch. By the time they roll one out, the next one is already in assembly. In a pinch, a dragon cargo capsule can be fitted with straps and foam, loaded with PLSS packs, and on the pad in about 2 days (It takes a day to move the crawler each way). Finishing up the rocket could be a week or two. Current production is a Falcon 9 every month... (cite).
Rescue by Russians or Chinese
The Soyuz program launches about once every two months, with the capability to push to one every 3 weeks, as I recall. So, a Progress with seats installed could be sent up for emergency rescue in probably a month, along with extra PLSS packs and suits.
The Chinese are much slower - about 1 per quarter is what I've seen guessed for them. In an emergency on the ISS, they are quite likely to sent a ship within a window of 3 weeks, if they have a launcher far enough along.
Since the scenario includes no soyuz available at station, and immediate extraction the failure mode is "death."
The best possibility would be if a CRS mission was waiting to launch. in which case, unloading time, and charging time for the needed PLSS packs. Which would mean, assuming they simply write off the cargo, about 6 hours, leaving the cargo in the launch gantry.
If a Cargo mission or a personnel swap isn't scheduled, then "immediate" would be at least 2 days, if not a week or two, for the next available launcher.