The issue is your assumption that the horn and trumpet are in fact in F and B-flat. Trumpets can be pitched in a variety of keys, and horn historically has played in one harmonic series or another without the use of valves, but by using crooks to pitch the instrument in one key or another.
In this case, the horn part is written in C and the trumpet part is written in A. Modern-day players reading this music would need to transpose to the key of the instrument they have, or use an instrument with the matching key. Modern-day horn players are VERY adept at this kind of transposition as the classic horn parts come in a wide variety of keys, and professional trumpet players will typically have a few different instruments on hand; but if the trumpet player doesn't have an A trumpet, he or she should be comfortable doing the half-step transposition to B-flat.
More likely in this particular case, however, is that only the public-domain score you are looking at has the odd-transposition brass parts -- the individual instrument parts you'd rent to perform The Mikado would almost surely be available in the more typical transpositions of F and B-Flat.
The syllables of Ut and La here come from traditional fixed-do solfege as seen historically in European classical music. Wikipedia
The key signature issue can be explained as follows: The key of the piece is C Major (or A minor -- but that's moot here), so for the trumpet playing in A to sound up a minor third in C (sounding), the written key needs to move from C up a minor third to Eb, which is a key signature of 3 flats.