Author Rachel Ehrenfeld recently won the right to seek relief in U. S. Federal court against the British judgement against her, on the grounds the British judgement violates U. S. law. Other incidents in which American authors were affected by actions in British courts are also related in the article linked above.
Libel tourist use the British courts because Britain has a huge hole in their system of laws. There is no formal right to free speech in the British system of justice. It wouldn't solve every problem, but it would certainly help.
Formalize the right to free speech, and tighten the requirements for bringing a libel action for what somebody writes. The positive, definitive establishment of truth as an absolute defense would help as well.
But, in the long run the British establishment needs to grow a pair. British apparatchiks are just too ready to cave in to jihadist demands. The professional bitchers need to learn that they can't have everything their way. Your kid goes to a school where a teacher has a piggy bank on her desk, you have no right to demand the piggy bank be destroyed. And if I ever get a cat, I'm naming him Mohammed.
How valid the complaint is is what matters, not how loud the complaining is.