i see you are from London.I've been working without a car for 2 years and am learning driving at the moment. i live in NW London and travel almost every day to central London, as my rates are too high for local area, so I hardly have any clients locally.It depends what treatments are you going to offer. I started with just nails and a large silver case was enough for me. Now I am qualified in waxing and make-up, therefore I need a car, as I often get requests for combination of these treatments and it means more stuff to carry.
I would say, go for it to start off and then learn driving. Using taxi or minicab is too expensive. I only do it in emergency or for urgent appointments when the client is happy to pay for it. Plan your routes in advance and you won't waste time waiting for a bus.
Also, it depends on your target area - in Central London there is a major problem with parking, so consider that too.