Why don't you look at criminology? A law degree can be a bit dry as there are lots of less exciting elements to it. And criminal law only takes up a very small part if most standard law degrees. That's my experience anyway, although it's well over 10 years since I did my law degree so things may have changed.
Are you looking for advice on studying or jobs? There are many jobs that you can do in the general criminal justice area, anything from legal secretary (with the option to work your way towards ILEX qualifications or similar) or you could work as part of the court staff - with entry level posts as Ushers, through to list office and court clerks. Or you could work as a barrister's clerk.
I would second the suggestion of studying criminology, perhaps combined with something like sociology, if you know you don't want to be a lawyer. Many FE places offer foundation degrees which you can progress to a BA. You could always convert to the legal qualifications later if you decide to do so.
If you actually want to work in the legal sector, I think qualifying as a legal executive through ILEX is quite a good bet these days. The traditional law route of law degree and LPC (the professional qualification for solicitors) or BPTC (for barristers) is insanely competitive - like, you need to have a high 2.1 or a 1st from a good uni and impeccable academics generally. Plus the LPC costs about £13k in London so unless you have a training contract already where your firm will pay, it is insanely expensive.
With ILEX I think you can work as you study and the option is there to qualify as a solicitor at some point in the future, just via a different route. TBH I think if you want to learn you might as well learn in a way that qualifies you for something and enables you to make money. You don't want to spend £££ on a qualification that doesn't actually qualify you for anything
The appeals process is a procedural thing, and you wouldn't learn about that very much in a law degree (apart from the general principle of appeals) and I don't think it would be covered at all in a Criminology degree (although not expert on this as I read law).
When you say you are interested in the appeals process, what do you mean? The appeals process is not separate to the rest of law, iyswim, it's just part of the general process. If you lose a case, you might appeal and it would be then heard by a higher court who would then overrule the previous court or agree with them. You don't get lawyers who just do appeals - it's generally a part of the job with some cases.
Don't be put off legal work because you're a bit shy - only barristers stand up in court and argue/cross-examine etc. Lots of lawyers are introverts. Most of legal work is to do with being very methodical and thorough, and you need to be pretty bright but genius isn't necessary.
But do be aware that studying law is not just about criminal law - if ILEX is like the LPC, a lot of it will be to do with company and property law as well (but it is interesting).
BTW I'm a trainee solicitor at the moment so am happy to answer any questions if I can.
As plutarch says a law degree is broader than criminal law. The appeals process is (very broadly) an assessment of whether or not the law has been applied correctly in a case - it is not a reconsideration of the facts of the case.
You can train on the job as a lawyer (as above) or you could do an access course and a law degree. If you want to be a lawyer via the law degree route then you need to take a certain range of subjects such as Criminal, Contract, Land etc for it to be a qualifying law degree so you can go straight on to law school for the vocational course. This is a more expensive option than training on the job although it might be slightly quicker.
If your specific interest is Criminal law then maybe the Birkbeck Certificate of Higher Ed in Criminology would be a place to start - that can lead into a degree course in Criminology or Law..