I'm a private ESL tutor in California. I've been teaching privately for a little over two years. I couple of months ago, I started a blog about teaching students privately (as well as grammar tips, etc.). My blog is at Teaching ESL to Adults (this is a different site than the one listed under my name below). I have some info there on advertising for new students (I use craigslist.org), some info on where to meet students (I meet my students in a coffee shop, and I highly recommend that you do not go to people's homes--with certain exceptions). I can't suggest how much you should charge in London, but checking other listings under "Lessons" in craigslist will tell you what the going rate is.
I keep two to three different ads running on craigslist at all times (as students come and go). The students I've gotten from these ads has allowed me to earn a modest living at tutoring. I highly recommend that when you write the ad, you think about the problems your student is having and how you can meet those needs. Most people list ads telling all about themselves and don't tell how they can help the student. I first address their potential problems, then how I can solve them, then I write a little about my qualifications.
Teaching ESL to Adults