You don't need to do work experience with the big corporate firms to find out that you want to be a corporate lawyer. During sixth form, I did a week with a full service firm in Derby, which was pretty small and allowed me to do each day for the week in a different department. I did corporate/commercial, criminal, family, personal injury and conveyancing. Whilst I appreciate that the corporate stuff was a lot smaller than the MC would ever touch, it's the bases of contract etc that are important.
It was absolutely invaluable. Going to a purely corporate firm wouldn't have given me the understanding of the different types of law and their individual merits. I learned so much from that week and when I started university, I was pretty clear on what I wanted to look into in terms of careers.
No university or law firm expects you to go to a magic circle law firm, and quite frankly, I doubt whether many entrance tutors will really know much about the big law firms or care - some of mine never practised as a solicitor in the UK or at all.
The person above apparently managed to get experience at Freshfields - to get this at your age you'll need contacts and/or a bulldozer attitude that will either work absolute wonders and gain respect or completely smack you in the face when lawyers remember you for being irritating and arrogant. I'd say it's not worth the risk or wasted time/effort for something that is unlikely to have much effect.
My advice - try a full service local law firm. It doesn't have to be big. Getting experience in more than one area of law in s smaller firm will look FAR better on your CV/personal statement than just sticking to one area you can't possibly know you'll want to do in 4 years' time at a large firm.
The glitzy name isn't everything in this situation.