Two kinds of pumps were used, generally in conjunction:
The lowest set was usually a bucket lift. A wrought iron rod worked inside the column of cast iron pipes. At the lower end of the rod were two semi-circular hinged flaps. On the downward stroke, into the water in the pipe, these opened, allowing water past them. At the lower end of the stroke, they fell shut, and on the upward stroke they lifted the water inside the pipe. Bucket lifts were preferred for the lowest set because they could cope with a certain amount of mud and debris without getting clogged. There would be a perforated strainer at the bottom of the pipes, but some mud debris always got past the strainer. At the top of the pipes was a large wooden box called a cistern which could hold the water lifted by a number of strokes of the pump.
At intervals in the pipes were hinged non-return valves called clack valves which stopped the water flowing back downwards.
The cistern was pumped by a forcing lift of pumps. At the bottom of the pipes entering the cistern was a large H-shaped cast pipe. One of the vertical uprights of the 'H' formed the bottom of the column of pipes in the shaft. A rod (wood or cast iron) worked a hollow casting up and down in the other vertical leg of the 'H', forcing water through the horizontal leg of the 'H' and into the column of pipes where it displaced the water already in the pipe further upwards. Clack valves prevented the water flowing back down. Forcing lifts were fine where there was little or no mud or debris, and so were used in the second and subsequent lifts.
Each lift could be 150 to 200 ft high. Each lift had a cistern at the top with another forcing lift working into it. At the very top, the pipes would discharge into a wooden box-shaped launder (channel) which directed the water into a horizontal tunnel leading to surface, the floor of which was inclined very gently towards the entrance, preventing the the water from flowing back to the shaft.
One pump rod usually ran the depth of the shaft, with offsets to work each lift. The weight of the rods was counterbalanced on surface by balance boxes attached to a second crank on the opposite side of the waterwheel to the main rod.
Motion of the rod was changed from horizontal (from water wheel, along ground at surface) to vertical (where it entered the shaft), and around any changes of inclination in the shaft, by angle bobs.
Many mining instruction books of the C19 cover the subject in detail. One with excellent engravings of pumps in shafts is J.Taylor "On pumps used in mines", Records of Mining, vol.1, London, 1829, pp.125-139 (facsimile reprint by Mining facsimiles, Sheffield, 1986.