I had a recollection that there was an early mail coach in the National Railway Museum
v quick google confirms that as per this page - looks like a replica of an 1838 example from the Grand Jct Railway built by the LMS in 1938:
en.wikipedia.org
Must be plenty more info on this example out there, not least presumably accessible via the NRM if not on line, plus ability to go an see it assuming it is on display still.
This looks helpful
Traveling Post Office, Grand Junction Railway, replica postal sorting van, copy of first TPO on London North Western Railway (LNWR) wagon frame, built 1938 (original 1838). Royal VR monogram painted on door sides, coach number 282693.
collection.sciencemuseumgroup.org.uk
also British Postal Museum might have other stuff.
I guess there were plenty of railways that carried post on non dedicated vehicles at that stage, as other have mentioned.