My impression is that European Circuses built their own building and did not travel from town to town. American Circuses were far more mobile. They built wagons and pitched big-top circus tents. They also travelled from town to town. There were some experiments with using railroads to move a circus, as early as 1851, but the first circus to move everywhere by train was the P T Barnum circus, in 1872. Barnum was an old man by 1872, and the actual manager of the travelling circus was William Cameron Coup. Barnum stayed in New York City and handled the publicity, but the logistical management was in Coup's hands.
Source: "The Circus moves by Rail" by Tom Parkinson and Charles Phillip Fox. This is the blurb on the back of the book. "400 pages of the most colorful trains and railroad equipment of all time, from tiny two car circuses to the giants of the industry. This giant circus filled volume is the greatest compilation of photos, posters, text and data ever compiled on the transportation systems that made modern circuses possible." Unfortunately, it only covers the United States and Canada. it has no information about any circuses trains in England