This has to be the world's lamest attempt at adapting any pre-existing work. The Cool Mikado is not an updating of the original Mikado's story; it has almost no story at all. It appears to be merely an excuse to get a few British comics some screen time to do their shtick, none of which has anything to do with anything that's been happening in the movie. Tommy Cooper comes off worst in this regard, spewing one-liners that are so unrelated to anything at hand that he sounds completely demented.
The movie shoehorns in a half-dozen songs from the original operetta, with lyrics completely unchanged to match the movie's situations. "We are gentlemen of Japan" is sung by a group of British soldiers milling around the cramped studio set. Ko-Ko is suddenly referred to as "the Executioner" so that they can sing "Behold the lord high executioner" for no apparent reason. Stubby Kaye does sing updated (but irrelevant to the movie) lyrics to "A more humane Mikado", which I suspect he may have provided himself since he is the only one involved in the production that seems to have the slightest inkling of how to make a movie. The most imaginative part of the whole movie is the "Titwillow Twist", and that only appears imaginative because of how dismal the rest of the proceedings are.
(The song most likely to be anticipated by any audience, and easiest to adapt to any production, is "I have a little list", but this is reduced to a throwaway one line joke. Incredible.)
Staging for the musical numbers wouldn't pass muster for a school play. It's all the players can do to just not trip over each other. The script wouldn't make a bit of sense to anyone not stoned out of his mind. And with bizarre situations appearing for no reason (such as a squad of bagpipers showing up for the final number) if you're not stoned out of your mind when you watch this movie, you may wish you were. Maybe that's the actual point of the movie--in its own claustrophobic, amateurish way, it's more psychedelic than most movies of the 60s that tried to be psychedelic.