To try and put in context. As long as the guy had a genuinely difficult/unhelpful time dealing with BA customer service then he's fine. If that isn't true, for example they responded quickly and repeatedly but he ignored it. He shouted and was rude to them immediately on the phone making it impossible for them to capably serve him etc. In those circumstances they would have valid grounds for libel in the UK because what he is saying isn't true. Though BA would never pursue it because it would simply make it a bigger story.
But the word "horrendous" is not a factual word; it has no legal meaning. BA can say "we helped him 15 times and found his bag in 20 minutes and paid him 100 pounds for his trouble", and those are all factual statements that they could (and should, if true) present for their side of the argument in court of public opinion. But "horrendous" is still an opinion and opinions are not libelous, in the United States. Like I said, in the UK things are different, and he probably can be sued for his statement of opinion--but if he is a US citizen then the US will refuse to extradite him for a free speech violation.