Diagnosis
The button itself is probably fine. This usually happens when the machine is still too hot from a long styling session—it won't give you cool air until the internal components drop below a certain temperature.
How to Fix It
Turn the machine off completely. Unplug it. Let it sit for a good 10-15 minutes to cool down. Plug it back in, turn it on to a low heat setting for 30 seconds, then press and hold the cool shot button. You should feel the air switch to cool. If it works now, you just needed to let the thermal regulator reset.
If That Didn't Work
If the cool shot never engages, even from a cold start, there's likely a fault with the thermistor or the button's connection. This requires a professional repair. Dyson's diagnostic tool can pinpoint it.
Heads Up
Don't hold the cool shot button for more than a minute straight. advises against prolonged use to prevent over-cooling the motor, which can also cause issues.