That would make it possible to record the stream. The UK legal system is very apprehensive of allowing any recording of what takes place in a courtroom that isn't filtered through a human observer's memory.
There is no objective public record of English trials.
Unsurprisingly if you request transcripts of a case in which someone was accused of raping a child, the witness statement from the child and associated examination by counsel may not be available to satisfy your "idle curiosity" whereas they might be available to the defence lawyer for the accused who is appealing the sentence based on something said in that part of the trial.
There is a general principle of the court publishes a fair, accurate and contemporaneous record except where this would create a substantial risk of prejudice. Here are the details https://www.judiciary.uk/wp-content/uploads/2015/07/reportin...
These UK reporting restrictions are also why the BBC (and other major UK-based media) are very careful about how they report on criminal proceedings.
Looks like the difference is that there's no jury to be prejudiced, because trials are decided by the judges only.
I wonder though if the extradition request is going to be decided by the jury at some point, in this case in particular.
It should be pretty obvious why.