Believe it or not, I was a London Bus driver once.
Official Transport for London policy is that animals are carried at the discretion of the driver/guard. This gives a 'get-out' clause should anyone want to bring on a large hungry Puma.
I believe the airlines are confused about transporting animals. Some airlines around the world were daft enough to transport animals in unpressurised holds. Of course, some died and the rest of them didn't want to get sued. As far as animls inside the cabin, well, I have heard it said because the air is recirculated, airlines think that fleas on animals may be infected with tropical diseases and they could get into the air system. So, like all good jobsworths' rather than find a definitive solution, they just ban animals from the cabin.
Can't think of a reason why Eurostar won't carry them. But like everything else, everything comes down to money. If any kind of transport provider carries animals, the public liability premium increases. And you can't legitimately distinguish between a soppy, half deaf, elderly Labrador and a fierce Bull Terrier with attitude.