What is a push bike?
For bicycles they do make/sell speedometers that use a magnetic sensor that is fixed to the frame along with a magnet that is attached to one of the wheels. The sensor send a pulse each time the magnet passes indicating one turn of the wheel. By knowing the circumference of the wheel and the counts per unit time the speed can be calculated. A small computer provides the calculations and dispays the output. Nashbar, among others, sell them. I am sure plenty of people have made their own. A PIC or some other small microcontroller can probably handle this task.