I love going to markets when I'm in London, and then I prefer east London. It's not always easy to find something there, but there are always some stands with small, independent, and often eco-friendly clothes. My favourites are Spitafields Market and Sunday Up Market. Go to Liverpool Street Station, from there it's just a short walk away! Best look it up on a map if you're going.
I went to London in March and asked for shopping suggestions, one of the recommendations was for Joules, and I ended up buying a dress there! Colourful, cute and fun pieces!
Hope you'll have a wonderful trip!