I agree with Hunka about the effectiveness of the film at conveying the details of what it takes to mount a production, and of its thematic connections to Leigh's other films:
Topsy-Turvy is nonetheless very much in the Leigh tradition of showing everyday work and frustration, even though there often doesn't seem to be much point beyond the ability to endure.
That theme -- that people do work for its own sake even though there is not "much point beyond the ability to endure" -- is reflected in Leigh's attitude to The Mikado itself, which he clearly dismisses as a featherweight bit of pointless escapism. Leigh has made it clear in interviews that part of what interested him about Gilbert was that he put so much care and effort into mounting such trivial stuff. And in Topsy-Turvy we see that everybody in the film is doing the best work they can do, not because of any great artistic ambition or belief that they will create something that will last, but just because it's their job. Gilbert writes The Mikado only because Sullivan rejected the script Gilbert really wanted to do; Sullivan would rather be doing grand opera; D'Oyly Carte is mostly concerned with keeping the team together and keeping his theatre profitable. It's a backstager that focuses entirely on the craft of theatre, as opposed to the art.
What makes the movie unsatisfying for me, ultimately, is that Leigh is so focused on this theme -- the theme of all the effort that goes into doing something trivial -- that he misses or downplays a lot of aspects of The Mikado and the period. For all the meticulous period details, Topsy-Turvy presents a pretty standard and very modern-day perspective on the Victorian period and the place of The Mikado in it: the Victorians are kind of smug, co-opting all cultures into their own, and The Mikado is an attempt to capitalize on the Victorians' arms-length interest in Japanese culture, a way of gawking at another culture without really having to learn anything about it. What Leigh forgets is that the big joke of The Mikado is that the surface trappings of Japan-mania, all the stuff that Gilbert saw at the Knightsbridge exhibition, are used for a story and characters that are 100% Victorian English. Instead of a look at another culture, it's a look in the mirror, with England's snobbery and sexual prudishness clothed in "funny" costumes. As in a lot of Gilbert's plays, the joke is on his own audience. None of that comes across in Topsy-Turvy, and the question of who exactly the play is making fun of never really seems to come up.
Now, a biopic doesn't need to be accurate about the thing it's portraying -- though in a film so determined to be accurate about other things, this is a bigger problem than in a different kind of biopic -- but Leigh's lack of interest in Gilbert the satirist ultimately makes him a less interesting character onscreen than he was in real life. It also fudges some of the more intriguing differences between Gilbert and Sullivan. The movie conveys their different lifestyles: Gilbert the repressed Victorian gentleman, Sullivan the high-living party animal. But it doesn't fully convey the irony of the fact that their ambitions were exact opposites of their lifestyles. Sullivan was a snob in the old sense of the word, meaning a social climber; he hobnobbed with the nobs and wanted to be the founder of an English national style of serious music (which, in his serious pieces, basically meant a whole lot of songs with titles like "Ho, Jolly Jenkin"). Gilbert was a commercial playwright, but the angriest commercial playwright in Britain before Shaw came along; his plays can be ferociously cynical, he could be corrosively funny about the stupidities of class pride and patriotism, and his favorite target was his audience's own bad taste in theatrical entertainment and culture. This repressed gentleman was in his own topsy-turvy way almost as angry as that other repressed 19th-century gentleman, Henrik Ibsen. Again, if Leigh knows this about Gilbert, he (and Jim Broadbent) doesn't get it across, and so we're left with a duller character than we should be getting. And while there's nothing wrong with taking liberties with a real character, there is something wrong with taking liberties that result in something much less enjoyable than the real thing.
Also, there's a structural problem inherent in Topsy-Turvy's focus on the rehearsals for The Mikado: the rehearsals were dominated by Gilbert, whose detail-by-detail staging of his plays helped to invent what we now think of as the craft of the stage director. Sullivan rarely got to do much more than rehearse the orchestra and the singers, and expressed frustration at being little more than a cypher at the rehearsals. So Topsy-Turvy, supposedly a Gilbert and Sullivan movie, becomes mostly a movie about Gilbert -- who has been drained of most of the elements that could have made him a character worth following for all that time.
In the end, Topsy-Turvy is a movie so focused on work for work's sake that it ignores what all that work was for. And in this context that says more about Mike Leigh than it does about The Mikado.