Its interesting as if it was an assistance dog, the law requires they take it. Pets are a whole other thing. Legally for assistance dogs the cab driver has to carry them unless they can show a medical certificate. However a lot of people and this includes many Muslims are actually frightened of dogs. Indeed some shitless. Now whether its due to the indoctrination of them about dogs, or whether its from experience I do not know. I might suggest you talk to a company who has done work for people with assistance dogs and ask if a driver would be able to be used for yourself. It seems silly if they are forced to do the jobs by legislation but cannot find the drivers they find for the disabled for yourself.
Most of the drivers I see when i travel with a friend with a dog, have a throw in the back and the dog has to sit in the footwell.
This is how assistance dogs are taught to travel, ie not on the seat with their heads hanging out of the window! Having said all of that however, the number of no shows of taxis for guide dog users is going up. This is probably because they know they would get a
1500 quid fine if they refused, to carry the dog and hence they see the dog from down the street and go right past. I feel there has to be collusion in this from the operator though but they deny it. Brian