Firstly, welcome to the forum.
I can't claim to know anything about a child setting up a business (I'm not being rude but in the eyes of the law, 15 years old means you're still a child) but I have set up my own business so know a bit about that side of things and these would be my musings regarding your position....
Banking
Can you open a junior/child's account with all the features required for business? ie; standing orders, direct debit, chip and pin card, online banking?
Tax & Self Assessment
At 15 can you register for taxes? Maybe speak to HMRC and find out how you register for self assessment.
Funding
Businesses aren't cheap to setup and run. A bank won't lend to a 15 year old so how else can you set yourself up? Can you get funding for computers, software etc... How about if something goes wrong with a job? How will you finance putting that right?
Insurance
Can someone under 18 get Professional Indemnity insurance?
Business Hours
Clients will want to talk to you and arrange meetings during business hours. How will you get round that when you cant take calls or meetings during the school day? With this in mind, how will you grow the business and target potential clients?
Experience & Knowledge
Many successful designers have no formal training and just as many successful freelancers have very little commercial experience. However, having no formal training AND no experience is going to put you at a massive disadvantage over the competition.
Time
New businesses require a huge time investment over a number of years. Do you really have time to invest in setting up a business? GCSE's are full time courses which you have no choice but to study for and running a business (no matter how small) will take time away from your studies.
Please don't think I'm being negative but at 15 years of age you've probably got more than enough going on without adding a small business in the mix. Why not continue to hone your skills over the next few years, try getting some work as an intern with an agency and maybe pick up the odd client of your own along the way. Then once you've got education out of the way, you'll be able to give the business and your clients your full attention.