There is no answer to the question, as presented. The criminal law differs in various respects in England, Scotland and Northern Ireland: one cannot speak of 'UK law', because criminal procedure is really very different in the three separate jurisdictions.
In England and Wales, the rules of criminal procedure are what govern the issue raised in the question, rather than the substantive laws relating to the various charges which might be brought; and while the substantive laws in England and Scotland are often fairly similar, the procedures are usually very dissimilar.
In a trial at the Old Bailey, in London, a Jury has no role in the criminal procedure: once empanelled it performs its usual function of deciding the facts of the case, based upon the evidence presented, but it has no role in determining what charges are preferred against the accused (this is decided by a different court at a much earlier procedural stage), and the jury has no power to alter the charges on the indictment.
Only the prosecution can decide what charges are made against an accused person. And if an amendment to the indictment is thought to be appropriate, only the prosecuting barrister can make such a change. If the case has come-on for trial, the permission of the Judge must usually be obtained to any alteration in the charges.
The Judge might object to an attempt to add a charge carrying a more severe penalty at a late stage in the proceedings (as the case may of course have already taken many months to reach the Old Bailey). He will often be more accommodating to an application to reduce the charges, to a lesser offence, particularly if accompanied by an undertaking not to proceed on a more serious charge - e.g. due to a lack of evidence supporting it.
Where several charges are brought in the alternative, then the jury has a function, since it can then convict of a lesser charge if the evidence on a more serious one does not satisfy it.
But it cannot ask for the charges to be altered: the jury represents the layman, and jurymen are inevitably not legally qualified (at one time, being legally qualified was an automatic disqualification from serving on a jury). So the jury is assumed to be incapable of understanding the fine distinctions between different offences, and has no role whatsoever in deciding which offences shall be included on the indictment.
Even the Barrister representing the accused has no role in determining which charges his client will face: that is purely a function of the Crown Prosecutor's office, and once the trial has come-on at the Bailey only the prosecuting Barrister and the Judge truly have a role in making any necessary amendments.
The function of defending Counsel is to strike a plea-bargain, if he can, and where the opportunity arises: which is to say, if he can persuade his client to plead guilty to a minor charge (whether or not on the indictment), and can also persuade the prosecutor not to proceed on the more serious charge(s) on the indictment.
A Judge will not usually object to a legally-represented defendant applying, by consent (i.e. through the prosecutor), to amend the indictment in order to enter a plea of guilty. He might refuse, if the accused has no legal representation, but not otherwise.
The short answer, therefore, is that in England and Wales a defendant cannot be convicted on a charge that is not listed on the indictment, but a jury can convict of any charge on the indictment if the judge asks the jury to retire and consider a verdict - but the case may not get so far as that, if a plea-bargain is struck.
There are rare occasions where a judge might withdraw a charge, if he desires, by directing the jury to acquit on that particular charge, but this only occurs if he considers that any conviction on that charge would be positively unsafe in all the circumstances of the case.
