The Mail Coach era began when John Palmer's experimental mail coach journey went from Bristol to London on 2nd August 1784, reaching London in 16 hours (something that had previously taken 38 hours). Other mail coach routes were then authorised by Pitt and by 1786 Palmer was made Surveyor and Comptroller General of the Post Office. Mail stopped going by coach with the advent of the Railways, the last mail coach running in 1846.
The Bath Chronicle of the period was filled with advertiesments for Coaches, as shown below.
The small map below shows the Bath to Bristol road, from Paterson's Direct Roads c. 1785.