Hi,
This is also a bone of contention between me and my mum! I've tried phoning our local taxi firms and asking them this, and they told me that they don't have car seats for children and that it's not their responsibility by law to provide them. They said that I can provide my own, but that most people don't bother. This shocked and confused me too..
What they suggested seems impractical to me as if I took my own I'd then have to lug it around with me wherever I'm going. And, like you, I don't feel happy about my nearly 2 year old not being in a car seat. So I haven't taken any taxis, which has limited me somewhat as I don't drive either.