We've been refused by minicab drivers with our dog, even though she was badly injured (torn leg ligament) and we were trying to get her to the emergency vet after hours. :\\\'(
Not sure what Black Cab policies are.
We've taken our dogs on the tube and train no probs and they are free.
When we moved out of London recently, we needed to get 2 small dogs, DH, me and toddler from West London to Kings Cross to get the train to the town we moved to in So. Cambrdigeshire. Having previously had the issues with the cabs, we needed to find a way to get them the 20 minute walk to the station (and the poodle was still recovering from the torn ligament, so walking them was out of the question, plus the streets are covered in broken glass anyway).
We used our extra pushchair, put dogs (toy poodle, westie) in car harnesses that we still have from the US, strapped them in and DH pushed them and I pushed DD. It was hysterical watching people's reactions (or lack thereof).
I think it would make a great candid-camera type gag. Anyway, worked beautifully for getting them to our new town. We then walked a mile or so from the station to the estate agent's office, then to our new house, and the girls did great. THe poodle just rested her chin on the bar across the front of the pushchair and the westie just rested hers on the side (one was in front of the other) or on the poodle. Where there's a will, there's a way.