It starts out with what might be described as a canon version of this video
with Azusa walking in on the others as they're playing (apparently) a Death Devil song. Didn't quite catch why. Mugi looks scary/hot with her hair covering her face as she plays.
By the way, the movie takes place some time during fall or early winter. It's cold enough for all the girls to wear both the yellow vest and jackets indoors. There are no leaves on the trees, but no snow. Minako Kotobuki gets something to do when Mugi tries to open the bag around a cake. As the others are discussing things, she's heard repeatedly "HNNNNNNN"-ing in the background for 30 seconds or so.
Sawako makes her first appearance in this scene:
As she joins in on the toast without having been in the room until then.
Then there's the opening credits with a lot of quick cameos, such as Yui's elderly neighbor and Satoshi. The whole movie has a LOT of cameos, as you'll see. Of the top of my head, the only even slightly regular appearing character who doesn't show up is the music store clerk. Opening theme is Ichiban Ippai.
Some of the classmates are discussing a trip to Hawaii (Himeko gets the first line), which gives HTT the idea of traveling somewhere (there might've been a discussion before this about the third-years wanting to do something for Azusa, I don't remember). They seem to want to to this as a surprise for Azusa, but she finds out about it almost immediately, so for some reason unclear to me, they move on to want to come up with something else to give to her. This comes up throughout the movie, with them trying to keep it a secret from Azusa.
They give their suggestions of locations, as seen in the trailer. Ritsu: Dubai (?), Mio: London, Yui: Europe, Mugi: An onsen (oh you), and Azusa being OK with anywhere. To decide on one choice, Yui draws a maze on a paper, and lets Ritsu choose a path, which ends up on Europe. Azusa then looks closer on the available goals and (surprise) all of them are Europe. This is why Yui has to wear this for punishment:
Also, after she gets up from her seat while wearing it, she manages to trip on her very first step.
The second plan is to put tea cups in the bottom of the aquarium, and let Ton-chan "choose" by touching (or possible resting in) it first.
Except Yui's Europe is represented by a tiny milk mug instead... At this point Azusa points out to Yui that London is located within Europe. After a long long time, it turns out to be London which makes Mio
very happy.
Then I think there was some discussion about what to do in London, and they visit a travel agent. At one point Yui sees a black cloak disappearing around a corner at school, and follows (with appropriate "sneaking around" music), and it turns out to be the girls from the Occult Club. Yui ask them if they want her to bring back a souvenir from England, and one of them respond that they would like a picture of Nessie, which causes them both to break character and start to giggle. Yui doesn't understand and instead says that she'll see what he can do, to their confusion. Unfortunately, this didn't seem to go anywhere...
Eventually the day of the trip comes, and Yui gets help from Ui to pack her suitcase. Well, that is until they're joined by...
their parents! At first their heads are hidden by the top of the frame, but then it cuts to Yui leaving the house, and they're standing fully visible next to Ui.
They join up at the train station, and go to the airport. There was something I liked about the background music here. Just simply good travel music, or so.
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On the plane we get some wonderful Aki Toyosaki English when Yui wants to know how good Azusa is at it (she refers to her as "Azu-cat"). Also, Azusa points out to Yui that different time zones doesn't mean that an airplane is a time machine.
And most importantly there's some sweet Ritsu bang service.
When they arrive at Heathrow, the background music is the regular tea-named BGM, except it's played on bagpipes and drums. Mio's luggage doesn't appear on the belt, which leads to Ritsu pointing out that this would mean that Mio won't have any pantsu to wear, but then it turns out to be standing in a corner of the hall.
The IMO funniest scene is when they go through immigration:
It's five quick cuts with the girls saying one or two words each (like, "Hello"), and finally Azusa, whose line is "Yes. I am seventeen."
They then take a taxi, which means the first English speaking character. According to the credits, there were a bunch of English-speaking actors playing the parts, so no broken English, and while I'm no expert on various British accents the cab driver did sound very British.
Oh, and Yui has trouble opening the passenger door when the handle isn't where she expects it to be.
They come to the hotel, only for the receptionist to tell them that they don't have a reservation there, it's for another hotel of the same chain at Earls Court. D'oh! So they take a bus there (well, Yui suggests it, I don't think it was shown).
At this point, either on their way to the correct hotel or after they've checked in, they're getting hungry, and notice a sign for a sushi restaurant ('cause when you're on a short trip abroad, you want to eat food from your own country?) They enter the place (commenting on how it's somewhat different from a Japanese equivalent), and are greeted by the owner who starts to talk to Yui in English. Yui is so nervous that she only says "Yes" to everything, including "You're the band that's playing tonight, right?" So before they know it, they've been given some robes to wear, and their guitars and bass are put up on stage next to a drum set. At this point they realize what's going on, and Mugi walks up to the stage workers. "Excuse me", before it cuts back to the rest (so sadly, no more Mugi English), relieved that she will fix things, but horrified when they see that what she asked them was to bring out a keyboard as well! Ritsu tries to tell the owner that they're customers, not performers, but he just stands there looking at her, so she starts stammering before saying "I like sushi?"
Here it can be noted that Ritsu is the worst English speaker of the bunch. She uses a lot of Japanese words, and almost never understand what anyone says to her until someone else translates it for her.
So they end up performing, playing Curry Nochi Rice (although it's a new recording of it). Afterwards, the guests applaud (with one of them saying "Arigato gozaimashita" with an English accent) and the owner says "Thank you, Love Crysis", and they end up sitting on the street outside. Still hungry, since they feel it would be embarrassing to walk back in and order food after playing. But wait, Yui (or possibly Mugi) says, isn't Love Crysis the name of Ritsu's friend's band? No no, they're Love Crysis in Japan, the band booked for the restaurant must've been a British Love Crysis she points out. Except that someone than calls out for Ritsu (who stammers some English, until Mio points out that they spoke Japanese), and it indeed turns out to be the band from the live house. They talk for a while before the older girls go in to do their gig.
Come to think of it, I wonder if HTT got paid for that performance...
Then there's a montage of various things they do in London, set to Unmei wa Endless.
I'm not sure if it's meant this way, but when they cross Abbey Road, it looks like they're not aware of where they are. Azusa is staring at the map, and no one - not even Mio, who brought it up during the planning phase - acts like it's something special.
Also there's a clip of Mio running away from some exhibition in panic, I don't remember the name, but I think it was something about medieval prisons, like the Tower of London or something.
At the hotel, Azusa and Yui shares a room, and the rest have the room next door. And for some reason there's a door connecting the two directly. This results in a segment where Yui spends time in the other room (discussing their secret plans) while Azusa is sleeping. She then goes back through the direct door, just as Azusa has woken up and walked out in the hallway looking for Yui, and knocking on their door. When she hears that Yui just went back, she also uses that door, just as Yui knocks, panicking about Azusa not being in her bed, etc. It eventually ends with Yui rushing in to the room while Azusa is still there, grabbing her and exclaiming "Azunyan is gone, Azunyan!"
Oh, and I forgot, there's also a minor somewhat running gag about the girls being short. When they're about to start playing at the restaurant, the camera pans up on Yui from the feet as she's looking all cool, only to show that the microphone is like two feet above her head. And when Azusa knocks on the hotel door, Ritsu has to struggle for several seconds to reach the peephole.
Later they get a call from Janice (with Christina happening to sit next to her at the time, with a single line). Not sure what's discussed in this scene compared to the discussion with Love Crysis earlier, but they end up getting booked to perform at a small Japanese fair being held at a park.
Also, for a really obscure cameo, Black Frill, the goth loli duo, is also set to appear there. When they are about to start their sound check, Yui becomes afraid of plugging into the speaker. There was a scene at the hotel earlier where she was going to use a hair dryer/curling iron/whatever, and it short circuited when she plugged it in, and she doesn't want to hurt Giitah. But as she's hesitating, a black-clad figure hurries to the stage, grabs the cord and plugs it in. It's Sawako! Who's traveled to England for... some reason or another...
Naturally, Sawako has brought costumes for them to wear (which includes katanas), showing photos of Madoka, Ui and Jun wearing them, but they refuse.
They end up playing Fuwa Fuwa Time and Gohan wo Okazu. Earlier they sat down and tried to translate the latter into English, but Yui starts to sing in Japanese anyway. She does add some lines about London towards the end, before shouting "Sky high!" (which Ritsu wonders about afterwards). They then have to hurry to get to the plane in time, since playing at the fair wasn't in their original plans. It starts to snow.
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Back in Japan, Yui shows Madoka what she brought back: enough tea to almost fill their living room table.
Later they ask Sawako for permission to give a performance in the classroom, which she approves of. That is, until the disapproving teacher from the end of the second season walks in, and she quickly has to change her tune (while writing down and showing her approval).
A very quick flashback reveals that this teacher worked at the school during Death Devil's days, and were against them playing in the classroom.
They perform the next day, with the classmates pushing all the desks to the front, creating a stage like in the second season opening credits. They first play Sumidare 20 Love (I think, it was a Mio-song from the second album at least), followed by U&I. And unlike all the previous songs in the movie, this one is as far as I can remember shown completely uncut. During their performance more and more students from all around the school hears about the concert and goes to check it out, including the male teacher who slowly walks towards the classroom (with Sawako dragging behind trying to stop him), except it turns out he's okay with it. Awwww...
Oh, and Himeko is seen talking to Madoka, giving her lines in three separate scenes, more than any other classmate. Yui ends up jumping off the stage and stands in the crowd while playing and singing.
Later Mugi tells the other third years that she's come up with a melody for their song, and they listen to it on her headphones, not letting the viewer hear. They're also shown sitting at the usual burger joint suggesting lyrics. And then Yui is shown practicing, and it turns out to be Tenshi ni Fureta yo that they're writing. There's a funny scene when Yui is trying to write lyrics in her room, and Ui knocks on the door. The Yui-logic way of handling that? Hiding the note book while saying "Just a minute",
and pretending to be asleep when Ui opens the door.
Somewhere here. there was also a scene where Azusa and Jun played table tennis, but I didn't catch much of the dialogue there. I think it was about graduation and/or sempais. And a scene at Ritsu's place, giving Satoshi his mandatory line.
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It then somewhat surprisingly cuts to after graduation.
In the graduation episode, Mio, Mugi, Ritsu and Yui walks up to the club room, although Yui stops to talk to Madoka, before running up the stairs, and their next scene is when Azusa enters the club room. The movie shows what happens in between. When Yui gets upstairs, she sees that the others have walked out on the roof (apparently the door on the opposite side), and they stand/run out there talking about graduation. Yui then sees a dove flying, and it reminds her about how the others played Tsubasa wo Kudasai for her in the first episode. It also makes her realize what's missing from the lyrics that she hasn't gotten just right: the word angel (although since they were planning on playing the song for Azusa, I'm not sure what word they had been using up until then). One quick rewrite later, and the performance is shown again.
Except that it's not only newly animated, but most of the song time goes to show flashbacks of them practicing (and hiding it from Azusa), timed to the song. There's one really cute scene when they're sitting in Yui's room and she puts a stuffed animal in the chair to represent where Azusa will be.
After the song's over, Azusa applauds, and says how it wasn't very good, except this time the POV is from outside the building, showing Azusa in one window and Yui's reaction in another (and, being from the outside, the line isn't heard).
It then shows the girls walking home afterwards. I thought this was in the anime, but maybe I was wrong. It's basically the scene from the manga with them talking about graduating (most of the scene only shows them from the waist down as they're walking), except Yui then asks where they should go next year, when it's Azusa's turn to do a graduation trip.
THE END
The ending credits are a shortened version of Singing, with the "music video" having a British theme. The girls are shown walking along the cliffs of Dover, and some scenes where they are wearing the Alice in Wonderland-esque dresses, both as seen on the CD covers (Yui looks really cute in pigtails and big glasses, and it includes more Ritsu bangs). This is then followed by Fuwa Fuwa Time as the credits continues over the black screen.