If you've seen the brilliant musical The Book of Mormon, then Meet the Mormons (Channel 4) will ring a few bells. Maybe bring some of the songs back into your head too. "I believe that God lives on a planet called Kolob/ I believe that Jesus has his own planet as well/ And I believe that the Garden of Eden was in Jackson County, Missouri …"

Ha! It is actually the only musical I've ever enjoyed, I think. (Obviously those aren't the lyrics that are most coming back, but this is a family newspaper, and you might not want a song that's basically a call to have sex with God, in all his orifices, including his vagina, and his eye, over breakfast.)

So Josh, this dark-suited, neatly cropped young missionary in Lynn Alleway's film, is off not to Uganda but to deepest, darkest Leeds in order to bother the locals and fail miserably in his attempts to convert them to the Church of the Latter Day Saints. And it's not wicked Suárez-bitey satire, but documentary – sensitive, thoughtful, fair, objective (mostly, see later) documentary. Otherwise the story is pretty much the same.

Alleway has gained unprecedented access to the church. And to Josh, who looks like something between a young John Travolta and Rick Astley, as he sets out on his mission. There are a few days of sales training ... sorry, saving souls, not selling, at a centre in Preston, before Josh sets out to spread the word. Actually, not Josh any more; from now, even though he's only 20, he's Elder Field. Which sounds quite good actually, appropriately pastoral. They must have been an Elder Berry at some time too, no? Elder Flowers?

More serious than dropping his name, he also has to drop his family and friends for two years! A phone call home every six months, one letter a week – you can stay more in touch with people if you're in prison. But an Elder must concentrate fully on his mission.

When training is complete, he's teamed up with Elder Bauman, who is Swiss and will be his companion for the next two years. They're not actually shackled together, but frankly they might as well be. They can go to the lav separately, otherwise they must do everything together: eat, sleep, pray. And try to convert the people of Leeds. Who of course are not interested, or they're Muslims, or Sikhs.

It is funny because of what Mormons believe in: that Jesus went to America after his crucifixion, the planet Kolob, Joseph Smith, the secrets, the angels, the little white handbook, the not-at-all-little white underwear, the ban on swimming etc. Plus, Alleway gets endless lols from the creepy church representative – sometimes Des, sometimes Richard, but always there, outside the door, listening in to make sure everything stays above board and her questions don't get inappropriate. (She does get into trouble for asking about sex and celibacy.)

It's also dead sad. Because Josh ... sorry Elder Field would be such a likable, normal boy if he hadn't been indoctrinated with such nonsense, and if he was just a little more questioning. There are doubts there, clearly. He would have given up and gone home, he admits, if Elder Bauman wasn't there the whole time, watching over him. Elder Bauman, further gone, is both less likable and less normal – a bit creepy too, to be honest.

Earlier, at the training centre, Alleway notices Josh ... sorry Elder Field, has been crying. She talks to one of the officials. "Josh is feeling … sorry, Elder Field is feeling pretty sad today," she says. "Couldn't he be a good missionary and still be in touch with his mum?" And there's a wobble in her own voice now. The answer, of course, is no, as the official laughs it off. "He's just going through a normal period of adjustment," he says. "Contact can be worse than no contact." Really?

"I'd really like to give you a hug, but I'm not allowed to," she tells Josh (yeah, screw you Church of Mormon, I can call him whatever I like, and he certainly seems more like a Josh than an Elder at the moment, with his eyes full of tears). Not just his documenter now then, but in the absence of his real mum, she's taken on a little bit of that role as well. Perhaps this breaks some conventions of documentary making, but the film is all the more touching and human for it.

Does Alleway have a son herself, I wonder? The answer's on her website, the last line of the biography section: "Currently, I live in south London with my finest-ever production, my 22-year-old son, Josh."