This is a very well rounded application, the kind that would be enough to at least get to the interview stage in Oxbridge, dependent upon the LNAT, and your personal statement. The one thing they might ask about is the 7 you got at GCSE, but given your exemplary grades in all other subjects that shouldn't be an issue. In terms of building more experience, there are some online programs that are really great. Cambridge offers a free online course that is amazing for exploring the 3 main aspects of law (Criminal, Civil, Administrative). If you want to experience a bit of contract law, Womble Bond Dickinson has a free online course on its website (it's a challenge, but enlightening).
Put simply, this is an academic record worth applying with - if you want to study law at Oxford and / or UCL / LSE, go for it!
P.S. my tip for LNAT is just practice - there are no tricks for the multiple choice, you just have to get used to the different way of thinking, for the essay section, the key thing to do is form an opinion on everything, and have a logical, reasoned way of arguing it.