Thank you for taking the time to provide such an informed and helpful response.No answer from me on this, but there is a bit of background I could turn up, but it is a can of worms.
This Danger Man series was created I believe by ITC which was owned by Sir Lew Grade and they also produced many seminal series such as
The Adventures of Robin Hood
The Adventures Of Sir Lancelot
The Champions (1968)
Jesus of Nazareth
Man in a Suitcase
The Muppet Show
Randall and Hopkirk (Deceased)
This is just a short list of ITC productions during that period.
As for the instrumentalists on these soundtracks they were I believe session musicians and largely unrecorded regarding credits for their work and may have rotated around the work according to availability much like Uan Rasey in MGM rotated around the work although Uan is personally credited for work on many premier productions such as Chinatown, his work was however largely uncredited being simply an available session musician
Among Uans many credits were as you probably already know playing on soundrack for
An American in Paris
Bye Bye Birdie
How the West Was Won
My Fair Lady
Singin' in the Rain
West Side Story
I dont hold out much hope for tracking down the trumpet player appearing on the Danger Man sound track.
One possibility is cross linking other productions at the time in the hopes of coming up with a name or names of session musicians appearing on ITC productions.
Lew Grade owned talent agencies and theatre companies at the time however so the task of tracking the player down who appeared in just one series he produced is made more difficult. He could have been engaged for just one gig, we cannot know this.
The fact that it is not documented suggests it was not a prominent player like Roy Castle or Kenny Ball.
That was a lot of words to say "I dont know" but I hope at least some of it might help in your search.