In England it's called a horse and trap. The horse would generally proceed at a walk or a trot, so the speed would not be too high - around 8-10 mph at most, I should imagine. The following articles would tend to bear out that this was the sort of speed.
'John Cameron, employed by the Earl of Seafield, was caught driving what was called a light locomotive in Aberlour at a speed of 12mph. He was fined %A35, which would represent a sum considerably more when translated to present day values.
Apart from wondering what a 'light locomotive' looked like, how did they estimate the speed of the offending vehicle in those days? There would be no unmarked police cars in 1902. They certainly would not have had any speed cameras or hand-held 'guns' at that time. The speed must, at best have been an estimate, comparison with the speed of a horse and trap maybe?'
https://en.wikipedia.org/wiki/Locomotives_on_Highways_Act_1896
On the 21st February 1921, He was riding his bicycle along a country lane at Chesham when he saw a runaway horse and trap, with a lady occupant, racing along the lane. He pursued it for over 500 yards on his bicycle, and after the third attempt brought the horse to a stop.
http://www.mkheritage.org.uk/bch/docs/gallantry.html
In those days bikes would probably have an effective top speed of 15-20 mph, and this proved sufficient to catch a runaway horse and trap.
