How difficult is it to secure a city training contract in a prestigious law firm?

LLBhaver - I'm going to slightly disagree with you here. Agreed on ensuring the OP enjoys and gets stuck into his degree and that lots can change between now and third year etc... but uni is *also* the time to be filling out that CV with all the good stuff that law firms are looking for - irrespective of whether that's US, MC, wider city, national/regional etc... Those years roll by very quickly and thinking about life plans now (even if they change) has value.
Eddie - I'd encourage you to think about why you're interested in the firms your interested in? Why US for example over MC or other UK city firms? What are the differences? the advantages? the drawbacks?... What does corporate or banking work actually consist of at this practices? (it is very different to what most of consider "law" is). Also note there's not really a strict law firm "hierarchy of difficulty/prestige" with US firms at the top, then the rest of London, then the regions etc... They'll be strong candidates applying to all sorts of practices, competition can be just as tough in the regions for example because of the much smaller numbers of corporate/commercial TCs available.
Aiming for a TC is achievable at a large city practice for you, there's always a little bit of luck involved in exactly where you might land a contract (a friend applied to many city firms and received an offer from Herbert Smith and nowhere else). Generally speaking to have a good chance against the competition you'll want:
1) - A good 2.1 degree, a 1st can be helpful but is not a requirement apart from maybe some of the stuffier US firms. Warwick's a great uni that more than ticks the "well known/academic uni" box.
2) - Strong A levels (typically AAB or better) - assuming to get into Warwick for law you have that ticked already.
3) - Vac schemes and open days. Apply, apply, apply to as many as you can, they are competitive to and it's a bit of a numbers game. Have a couple of vacation schemes, the more the merrier, at similar(ish) practices to what you are targeting is very important.
4) - Other legal work experience - this is where you can really make a start now and try and lock in some solid work experience during your first year holidays. It can be whatever you can get your hands on, are there any small solicitors offices local to you where you could shadow for a week? Do any of your friends and family work in an office with a in-house legal department you could shadow?
5) - Something else interesting on CV to help you stand out and flesh you out as a person, a sport, a hobby, doesn't matter exactly what it is - but something.
All of this will stand you in very good stead for success at a variety of firms. The work exp/vac scheme side of things is really important, academics alone arn't enough on their own.