Are there any cases (from any jurisdiction) in which a successful litigant complained of bias? As in, even while the decision below was favourable to them, they are still aggrieved that a particular judge did not recuse themselves.
While the decision could not be appealed if the outcome was everything the litigant sought, the litigant (or anyone else who observed the lack of recusal) could grieve the judge with judicial regulatory officials for engaging in unethical conduct unbecoming a judge.
Given the lack of consequences to the litigants, the sanction for this misconduct would probably be very mild (perhaps a private reprimand that might aggravate punishment in the event of a future substantiated grievance), but this would really be the only options for action since suing the judge is not allowed (judges have absolute immunity from civil liability for their official acts), and an appeal of a favorable ruling is futile.
There is substantive reason that a successful litigant might feel aggrieved and file such a complaint.
While the successful litigant couldn't appeal the decision based upon the lack of recusal, the unsuccessful opposing party might very well be able to file a motion to have the judgment in favor of the successful litigant set aside and re-litigated, if the issue is discovered by the opposing party quickly enough. This would cause the successful litigant to incur additional expense and risk to maintain the same result that would not have been incurred if the judge had recused and another judge had handled the same case with the same result.
This uncertainty, even if the opposing party never actually takes action to set aside the favorable judgment in court within the allowed time period, could force the successful party to delay taking action in reliance on the successful result. It could cause the successful party to incur additional attorney fees monitoring the situation. Or, it could cause the successful litigant to reach a post-judgment settlement of the case for less than the fully successful outcome achieved, in order to definitively end this uncertainty. None of these issues could be raised on appeal by the successful litigant. But it would make the act of failing to recuse seem a lot less innocent and harmless in the context of a grievance against the judge.
Note also that this could happen even if the judge should of recused due to an appearance of hostility towards the successful litigant (e.g. the judge is the successful litigant's lawyers ex-spouse with whom there was a public and ugly divorce), and not just in a case where the successful litigant or that litigant's lawyer would appear to be too friendly with each other (e.g. they were best friends and roommates in college and were former law partners of each other).
If the successful litigant would be presumed to have been in a hostile relationship with the judge, the opposing party could argue that the judge bent over backward to rule in that party's favor and against the opposing party precisely to avoid the appearance that their hostile relationship had tainted the outcome.
When a judge decides a case there will be
An order - e.g. "Smith shall pay Jones £100,000"
Reasons for the decision - i.e. a description of the evidence, and the judge's findings of fact and legal reasoning.
Sometimes a party, even though they have completely won, is nevertheless aggrieved by some things the judge has said in the Reasons (or in the way the judge has handled the trial - e.g. the judge's interventions). For example the judge may have said that the winning party was not a credible witness but they nevertheless won because of the evidence of other witnesses who were found to be credible.
The rule in England and Wales is that you can only appeal orders. So if the order is completely in your favour you cannot appeal just because you don't like the reasons.