Edit: just double checked the rules for the ICE (regional fast train), if it fits in a carrier it's free of charge, a bigger dog you have to buy a child ticket for.

https://www.seat61.com/dogs-by-train.htm

> This bit is easy. You can take dogs, cats and other small animals with you free of charge on all British trains, up to a maximum of two per passenger, as long as they do not endanger or inconvenience passengers or staff. Dogs must be kept on a lead at all times unless contained in a basket. Dogs without leads, cats, birds and small animals must be carried in an enclosed basket, cage or pet carrier which must be rigid and not open (to prevent escape) and the animal able to stand and lie down in comfort. Animals and containers must not occupy seats, otherwise a charge may be made. If you want to use a Caledonian Sleeper to or from Scotland, you'll need to pay a fee, see here. For full details of dog & pet regulations on British trains see www.nationalrail.co.uk/passenger services/luggage animals.html.