The ’2 Men’ are David Steele (b. September 8, 1960) and Andy Cox (b.January 25, 1956), both at the time members of Fine Young Cannibals. They first recorded together as members of Birmingham (UK) ska band The Beat (2).
The ’Trumpet’ is Graeme Hamilton.
Note that Two Men And A Drum Machine exist as a separate artist (without A Trumpet).

The original name for the band was going to be The Untouchables, and was promo’d as such on LONDJ 141, but was dropped before official commercial release.