I’ve heard it said that various troubling decisions by British courts would set precedent binding upon many English speaking jurisdictions all around the world. I suppose this is in common law/commonwealth jurisdictions but surely there must be a fork in their respective legal cannons upon obtaining independence from the U.K.? How does this exactly work with many of these states having their own parliaments?
Contemporary decisions of domestic British courts (e.g. the Supreme Court) are not binding in foreign jurisdictions. They may be persuasive for courts there, because there is a common legal tradition and the U.K. judiciary is often considered to be quite good.
Some countries have preserved a right of appeal to the Judicial Committee of the Privy Council (aka "The Queen in Council"). This body is basically the same judges as the Supreme Court but sitting in a different room. As the JCPC their decisions are precedential in wherever the appeal came from, and potentially elsewhere if the law in question is the same. Corollary: JCPC decisions are not precedential for British courts, unless the JCPC goes out of its way to say so. In these cases, the JCPC applies the law and constitution of the origin country, which can include retained pre-independence British laws as well as the common law. That country could pass a subsequent statute or constitutional act overriding the JCPC's decision (as Parliament could in the U.K.), so that country retains ultimate control even though one of "its" courts is a shared one sitting in London.
This arrangement can cause dissonance. For example, some Caribbean countries would like to execute people, but that is prohibited in the U.K.; the JCPC continues to permit executions to go ahead in some cases, even though that would be illegal if the same judges were in a different room hearing a British case. There are very occasional cases from Brunei, whose law is determined by the Sultan and includes aspects of Islamic law not known in the U.K. legal tradition.