In my experience the English Small Claims Court system works exceptionally well, and usually without actually involving the court.
Given that the case appears to be quite straitforward ( It is their website - they are responsible for any errors or inconsistencies it might contain), prepare all the documents including the court submission and send photocopies to the airline, giving them an appropriate time in which to pay up.
You only need to involve the court if the other party fails to pay, which I have found seldom happens.