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.