This is the first time I've read about a input delay, and I followed several forums about NFS games, including Shift 2 for several years. I use a pair of USB joysticks setup to appear to be a single virtual device, and I'm not having a delay issue.
One way to test for this would be using nitrous, since the nitrous meter will start going down as soon as the game starts the nitrous. If you use nitrous, about how long is the delay? You could also use the brakes if this results in the tire squealing. You want to use a control where you can see or hear the game responding. If only the steering is being delayed, it's possible that it's a graphics issue and not a controller delay issue. In some NFS games, there's a bit of a steering delay depending on the car, because the physics has the car drift into a bit of oversteer before it starts turning, but I don't recall any cars doing this with Shift 2.
You could try reducing graphic settings and resolution to see if this has any affect on the steering input delay. It shouldn't have an affect on brakes or nitrous.