Buses will normally accept dogs for free, at the driver's discretion, if the bus is not too crowded. Once, i tried to get my rain-soaked pooch and i on the bus and the driver wouldnt accept us because he didnt want the dog to shake water all over the place.
You dont pay extra for animals on the tube.
Taxis ... it would be at the driver's discretion, i reckon.
When my dog had an operation and i needed to get him home without a car... i called like 5 different taxi companies, and none of them would take him. I advised he would be woozy and sleepy after his operation, and there would be no danger, but they all said no. Not sure what the issue was... hair on the seat? Peeing on the seat? Fear of being bitten? I really dont know.
I imagine a cat in a cage would be a different story, though... that would be just like luggage!
I bought a car soon after that nightmare!