The Trumpet is not the proper name of the pub, but a 'nickname'. It's real name is 'The Royal Exchange'.
It's history dates back to 1830, at which time it was a domestic house and butcher's shop until the then Duke of Wellington granted to the householder (on payment of a fee), the right to sell ale and beer from his premises. As it was a butcher's shop, it became known as 'The Butcher's Arms'.
In later years it became known as 'The Royal Exchange'.
An 'Exchange' was a place where people who could not pay their taxes in cash, could 'exchange' their various products for their market value (or tax value) with His Majesty's Collector of Taxes, and so pay off their tax debts to His Majesty the King.
It is almost certain that there is a pub of this name in every English town or city.
How it acquired the name 'The Trumpet' is a rather 'naughty' story and cannot be revealed in a 'family' orientated Web page, but can be read on a poster in the pub itself (you have to visit them find out !).