As explained well in an article by Emily MacManus, I don't think it's an exaggeration to say that British laws on libel are a threat to free speech around the world. Because they permit frivolous cases that would be far too costly for most people to fight, they give a great deal of power to anyone who is annoyed enough and has the resources to pursue legal action there. Even the threat of such action may be sufficient to make individuals or publishing organizations censor themselves.
The linked article goes further into the peculiarities, which include the characteristics of 'no-win-no-fee' litigation and the broad understanding of who constitutes a 'publisher.' For instance, if you said something true but commercially harmful about a company on your website, they might go after you, your internet service provider, or the company that runs the server your site is on. Any of them might feel pressured into removing the statements, rather than face litigation. British law also holds that "every time the statement is downloaded or accessed it constitutes a fresh publication," which could produce especially absurd outcomes for a popular website.
An important first step could be requiring the party bringing the suit to prove that the allegations are untrue, before the court accepts the case. For instance, if I said that Rio Tinto was polluting a river somewhere with mercury, or that Suncor is emitting huge amounts of greenhouse gasses, they would have to prove the opposite in some sort of pre-trial hearing, before they could come after me. It might also make sense to limit which courts can hear a particular case, so as to prevent people from shopping around in random legal jurisdictions to find the one that gives them the strongest hand.
The article suggests that 'principled deep-pocketed litigants' might be able to produce some useful new precedents, limiting the damage the existing rules on libel and defamation could have on honest and open public discourse. For now, however, it suggests that "the reaction to libel remains: take it down, take it down quickly, take it down again. And libel tourism means that this habit is likely to spread."
One thing the article isn't clear on is what could happen to you if a British court rules against you, in your absence, and you simply ignore them. Perhaps someone with more legal knowledge could explain whether there is any chance of them coming after assets held in another jurisdiction.