Here's a 2002 law review article by MP Henderson that does a rundown of ineffective assistance of counsel standards and precedents in both the UK and US (and also details shortcomings and inconsistencies in both, which is the true focus of the article, though not particularly relevant to this Q&A). The short of the matter is that bdb484's summary of the basic standard for the US is essentially the same in the UK: both focus on the practical impact of the counsel, rather than the actual efficacy and skill of the counsel. Meaning that if the court concludes that the result of the trial would have been the same with an effective counsel, then the results of that trial will stand. This is usually phrased in legal parlance as a "miscarriage of justice" issue. This results in a few seemingly glaring absurdities being ignored when the evidence against the defendant is seen as too strong to defend against, like sleeping through trials.
The specific case detailing the standard test in the UK is R (Aston) v Nursing and Midwifery Council  EWHC 2368, and is essentially the same 2-part test as the Strickland test detailed in bdb484's answer: was the conduct outside the range of that of a reasonable counsel/advocate, and if so was it materially relevant enough to alter the verdict?
The article I linked at the start states (with several citations, for the quotes in particular, here omitted):
Courts in the United Kingdom are highly deferential to counsel's decisions. Also, there is a strong presumption that counsel's conduct will fall within the wide range of reasonable effective assistance. Additionally, courts in the United Kingdom will make allowances for the "distorting effect of hindsight". As the result of such discretion, "[c]onducting the trial without, or even contrary to, the instructions of the client, mere errors of judgment or even negligence, may not be sufficient to set up an inference of an unfair trial."
And it's that last bit that's relevant to your question: simply defying the instructions of the client as concerns which evidence to present is not necessarily (and is unlikely to be) a basis for a successful claim of ineffective assistance of counsel. What constitutes a "fair" trial is a fairly broad umbrella; it doesn't require perfect counsel, nor that all conceivable avenues of defense be explored. Both UK and US lawyers are expected to both abide by the instructions of their client, while also being granted the leeway to present the case in the manner they deem best. This is spelled out quite clearly in the Scottish case Anderson v HM Advocate, where the High Court spelled out four points for what constitutes (in)effective assistance of counsel. To quote Henderson's summary again (which again has many citations here omitted)
First, the accused has the right to a fair trial and to have his or her defense presented to the court. If counsel's conduct deprived the accused of those rights, a miscarriage of justice could result. Second, counsel must abide by the client's instructions, and not disregard those instructions. However, counsel must conduct the case as he or she thinks best. Third, counsel determines how the defense is presented, and the accused is bound by counsel's decision. Finally, counsel, not the accused, decides whether or not to attack the character of a Crown witness.
Again, it's the third point that largely crushes your hypothetical's chances of success on grounds of ineffective assistance of counsel. For it is the counsel that ultimately decides how to present the defense, which includes what evidence and testimony to seek to include, and this is a binding decision upon the accused. The counsel in your hypothetical may have decided that the value of the evidence was significantly weaker than the client thought, and this generally falls under the umbrella of reasonable discretion for the advocate. Only if you can show the act to not present the evidence was outside the behaviors of a reasonable (generic) advocate, and also that it materially impacted the verdict, could you expect to prevail.
As a bit of aside, there is something of an issue on whether or not there is a right to effective assistance of counsel in the UK. The right to a fair trial was really only expressly codified in The Human Rights Act of 1988, but that does not specify a right to effective counsel, only the right to a fair trial. A trial can be fair with no counsel at all. By contrast, in the US the 6th ammendment has been held to grant a right to both a fair trial and effective counsel in criminal matters, and the government must foot the bill to provide the accused with an attorney if requested/needed (the accused can waive this right, and there is a certain reasonableness where courts will eventually tell a defendant "oh well, figure it out yourself" if they keep rejecting their court appointed attorneys and demanding a new one). In the UK access to legal counsel is largely considered only available to those who can afford to pay for it, with the government fronting the bill only in exceptional cases (e.g. the courts deems it desirable to do so in the interests of justice). This right to a fair trial rather than a right to legal counsel seems to be why the UK currently places the emphasis on the practical impacts of the alleged ineffectiveness, rather than the alleged ineffectiveness itself. My understanding of US jurisprudence suggests it's rather more practical in justification: the courts don't want to do do-overs over procedural issues when the result is a fait accompli, as it consumes lots of time and resources and risks the general ravages of time on evidence, will, and memory.