Any US law purporting to enact a broad muzzling of the press in court cases would almost certainly be ruled to violate the First Amendment. The only such laws that have been found to pass Constitutional muster (to use a slightly worn cliché) are those that are narrowly tailored to compelling state interests, such as protecting the identities of rape victims and jurors.
In UK the court proceedings are open. But what is subject to legal action is publishing articles that might cause the jury to consider facts not presented in court. So in the UK only the court proceedings can be published*. An interview with a 'victim' alleging abuse extra-judicially would be banned. Most UK residents although desiring salacious gossip about legal proceedings accept that limiting press comment to the legal process is just. A balance between open justice and fair justice.
*During the judicial proceedings.