Just some quick thoughts.
I'm fairly certain those A-level grades (incidentally far better than mine) won't get you into LSE or UCL for a course as competitive as Law. Someone I know had exactly those grades this year and couldn't get into a RG for Law even through Clearing. They are A*AA courses. You could do an Access Course and aim to get Distinctions (not imperative, but helpful) all the way through and delay your application until the following academic year. Alternatively, you can re-sit your A-levels and aim for A*AA, but re-sits aren't looked upon as kindly in general. However, if you can show that you were able to raise your grades [or get Distinctions on an Access Course] whilst juggling other commitments, it may skew things in your favour (obv, this varies from department to department and also tutor to tutor).
A little about the unis themselves.
I know of two people who got offers from UCL in my Access Course (Philosophy and Arch&Anth); so in those departments at least, Access students are welcome.
LSE seemed more or less indifferent (YMMV). I was interested in PPE and a friend was interested in History. The responses we received weren't exactly encouraging, though in my case, my lack of mathematical grounding would appear to justify the lukewarm response I received. In my friend's case, it made less sense since the course was in Humanities.
Good luck!
It may be useful to email them with your situation.