Great Britain's National Rail network is genuinely generous with pets: under the National Rail Conditions of Travel, passengers can bring up to two dogs or other small domestic animals free of charge, with individual train operating companies able to set their own additional conditions on top of that baseline.
Dogs must stay on a lead for the entire journey, including on station platforms and concourses — not just once seated on the train. Smaller animals other than dogs need to travel in a fully enclosed basket or pet carrier, capped at 85 x 60 x 60cm. Animals aren't permitted on the seats under any circumstance, and — except for assistance dogs — aren't allowed in buffet or restaurant cars unless the specific train company says otherwise.
Sleeper services need advance contact with the train operator if you're bringing a pet into a cabin, since space and policy there differ from a standard seated carriage. Assistance dogs are exempt from several of these general restrictions and are accommodated throughout the train and station.