Can I just say, this is a really well written article. Great diagrams, explanations, and it gives both sides fairly well. Even for the BBC this is top class journalism.
According to Wikipedia [1], the four-engine 747 and A380 comprise over 20% of BA's fleet, way above the worldwide average. The percentage of four-engine aircraft on BA's long-haul routes are likely to be even higher, and it's those long-haul routes that the people mentioned in the article are worried the most about.