According to Ball and Sunderland (2002), most hackney coaches in the Regency era were four wheeled carriages frequently castoffs from gentlemen. Because of a monopoly on taxi services, there was little incentive for owners to modernize the vehicles and therefore most were dirty and in disrepair. In 1829, the lighter two wheel cabriolet began to be introduced in London and was especially popular in Paris.
And here is an unflattering description in the guidebook of hackney coaches…
A view of London, or, The stranger’s guide through the British Metropolis (1804)
Ball and Sunderland. 2002. An Economic History of London 1800-1914
For more history of the hackney:http://18thcand19thc.blogspot.com/2014/01/hackney-coaches.html