While becoming an appeal QC is a long process, there are a number registered within the UK, and you do have a choice. This means that you need to do a little homework to find the best possible barrister for your appeal, and for your personal needs as well. However, every QC has risen above their peers, so there is rarely a bad choice when picking your barrister from their group.
The first thing you will want to check is that they actually specialize in appeals. Many barristers change over their specialties from time to time or offer a number of different specialties. Some even keep a junior who has a different specialty, keeping themselves open to more cases. This can be useful if you are appealing something minor and it mostly just a matter or filing paperwork, but if you actually need to get a conviction overturned, you want to make sure that you are working with the best of the best.
Next you will want to see how many years of experience the barrister of your choice has under their belt. All QC's must practice for at least 10 years before they are able to take the silk, but some have been in practice for much longer than that. Anything more than 15 years should be sufficient to prove that they have the skills you need. New QC's are often extremely motivated, but they don't always have the experience that comes with holding the position for a longer period of time. Many feel that they can be distracted from their cases by their new allowances, and that they have not yet learned to properly work with their juniors.
Finally, you will want to make sure that they have a good track record when it comes to your specific type of appeals. Having more experience and winning more cases in a specific category will make them more likely to win your case. They may even be able to reference their previous cases to prevent mistakes from happening in yours.
Once you have looked at all of this information, you are ready to contract with an appeal QC and get your case taken before the courts. Keep in mind that they have been working on similar cases for years and will understand what you need better than you will. A little trust in your barrister will go a long way.