Le Pain Quotidien
Le Pain Quotidien (which translates to "the daily bread" in French) is a well known chain across London and is a reliable spot for simple vegan fare that's filling and moderately priced. There are some seasonal changes to the menu, but they always maintain several vegan friendly options, including desserts. There's always at least one muffin that's devoid of all animal products (I had the blueberry and loved it – it's very wholegrain farm muffin style though, so if you want a more sugary baked treat then look elsewhere). Items that are suitable for vegans are clearly marked on the menu. All of the breads, save one, are safe.
It's a good option for people who want a light meal and enjoy fresh ingredients in preparations that aren't over over the top (the avocado tartine, for instance, is little more than salted mashed avocado with lemon hummus, beans, seeds, and fresh dill, but is still very good).