I have been a full-time tutor for several years now and also employ other tutors within my business. I should say that I do price at the lower end of the price bracket for my subject/level etc, but would still consider £15/hr ridiculously low for tuition, especially if you are travelling.
When considering your price you need to consider not just contact time, but your actual working time. Costs may include preparation of resources/planning sessions, regular time to feedback to parents regarding progress, marking, travel time, book-keeping (you will be self-employed, so don't underestimate the time required to keep spreadsheets of income/expenditure for your tax return at the end of the year), petrol, wear and tear on the car, paper, ink, text books, CPD, enhanced DBS disclosure, advertising, other stationary, phone costs, internet use (esp. if you are tutoring online) and more.
To give you a rough estimate, I would say that only approximately 65% of my income is gross profit (so my net profit is even less). If you are charging only £15/hr, you could well end up in effect earning less than minimum wage. (I realise minimum wage is not really a "thing" if you are self-employed, but consider whether as a highly qualified professional you really want to earn less than someone stacking shelves in Tesco).
Prices in some areas of the country can seem high, but when you consider all the additional costs of running a tuition business, they are really not.
I would suggest though that even if you are an experienced classroom teacher, as an inexperienced tutor, start at the lower end of the normal price range until you get experience and recommendations and then work from there.