Welcome! :-)

I've got most of the basics working however I'm having trouble with the
> transpositions. They aren't working as I expected. I thought the
> transposition function would take care of putting the parts into the right
> key. But, all of the parts have a key signature of d minor (one flat) (no
> key signature transposition is happening)
>

This is because you transpose only the notes, not the key signature which
is defined in an area (\global) not seen by your \transpose command.
Anyway, I'd personally prefer to have the music in the \TrumpetOne etc.
variables in concert pitch and only transpose them for printing; and this
solves your problem automatically.

So, leave the TrumpetOne in concert pitch (and it's not necessarily to
create a voice in the music variable):

TrumpetOne = {
  \set Staff.instrumentName = #"Trumpet I in Bb" % I prefer not to have
this inside the music variable - for instance, in the printed parts I do
not want the instrument name to be printed in the staff (but in the header)
  \clef treble
    \relative c'' {
...
    }
  }
}

And then \transpose at the point where the actual score is defined:

%**********************************
> music = {
>   <<
>     \tag #'score \tag #'trpI  \new Staff \transpose bes c' << \global
> \TrumpetOne >>
>
    .....
>
  >>
> }
>

Best
Lukas
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