Matt,
The airbag system has two modes - the User Mode and the Diagnosis Mode. You switch back and forth between the two modes using the ignition key procedure you used. Switching to the Diagnosis Mode doesn't clear the light; it lets you identify the problem.
In the User Mode the airbag light simply tells you there is a malfunction. Normally the light comes on for 7 seconds when the ignition is first turned on, then goes out. If it blinks after the 7 seconds, as yours does, there is a fault. If it stays on steady, there is a more serious fault, such as an airbag or seat belt tensioner that has deployed.
In the Diagnosis Mode the blinking pattern tells you which component is at fault. There are over a dozen blinking patterns, with combinations of long and short on times. The best way to tell which component is faulty is with the Service Manual. I know that UK manuals are virtually impossible to find, but the US manual may help.