Higher ranked universities do like people who have proven themselves at lower ranked universities, though some would argue it is easier to get better grades there. My friend got into St. Andrews and another got into Warwick (though paid 25k just in tuition!), they were both international students, so paid more than most and they come from middle class backgrounds in their countries.
Concentrate on your placement and getting the best result you can in your degree and be realistic, LSE is one of the best universities in the world, not just the UK. University of London is a good place to look though, SOAS, QMUL, KCL, RHUL, and recently City University is joining the University of London which might be interesting.
Getting caught up in elitism and ranking might stifle your potential, so really analyse who you are and what you want to do, as for example, you might be really interested in fashion and want to work for brands like Adidas and then as you mature, you could go on to work for a brand like Hugo Boss and with this ambition you would want to study for the masters in strategic marketing at the London College of Fashion, which is a part of the University of the Arts, which is not ranked at all compared to LSE, but within the fashion industry it is the elite.
St. Andrews might be a good crack as it is tucked away up in Scotland, Surrey is good and in the top ten and Loughbrough, York, Bath and Bristol. You'll have a good shot at getting into a good university.
Make the most of your work placement as that will be just as, if not more important to show your potential.