Writing of the early 19th century Stephen Bates says "Those servants who were to appear in public were dressed in uniforms; knee breeches and dress-coats for the footmen, flaxen wigs and cocked hats for the coachmen, as if they had stepped straight out of the eighteenth-century." [i]

Although 'Jeames' [ii] was published in 1862 the front page gives an idea of a coachman's uniform at the time 'Jeames of Buckley-square' was published in Punch

References:

[i]Bates, Stephen - Two Nations: Britain in 1846. (London, Head of Zeus, 2014) p227
[ii] G. W Hunt, Jeames.  British Library Shelfmark(s): Music Collections H.1257.(1.)