Sandy-Wroe Inadmissible evidence apparent pre-trial would be excluded pre-trial, at pre-trial hearings, and not seen by the trial judge. Any which comes out during the trial itself can be dealt with as it is now with the jury present.The jury is told to ignore it (and why) or, if that isn't appropriate, the jury is discharged and a new trial ordered. The judge alone would apply the same principle and either ignore it or discharge himself.
As a matter of practice, counsel (and judges) are accustomed to assessing the merits of cases when they have read all the inadmissible evidence that comes with the admissible.Throughout counsel's career the first papers that they get usually have some inadmissible material.Sometimes that seems to be most of the bundle! They have been trained to approach the case that way, judging it on what is proved or not by what's left after they've ignored the inadmissible. Judges have, mostly, been counsel but anyway have worked in criminal law and are used to this.
The Court of Appeal can always allow an appeal where inadmissible evidence has been admitted, if the effect of it is that the verdict is rendered unsafe. That applies to jury trials and would apply to single judge cases. It might, however, be less likely to do so in 'borderline' cases involving a single judge because of what I've just said.The Court might say that it's confident that the judge ignored the inadmissible but not sure that a jury, however well-directed.did so.