The film is "The Long Day Closes" by Terence Davies. It's a British film from 1992, I believe. (It's a masterpiece, by the way, as is his earlier "Distant Voices, Still Lives".)
In the scene, "Tammy" begins to play while the young boy (sorry, don't remember his name, and, given that it's a Davies movie, I'm not sure it matters) swings on a railing beside the front steps of his home. The music then plays over a series of overhead tracking shots through a classroom and a service in a Catholic church. (Listen carefully, and you'll hear a snippet of dialogue from "Kind Hearts and Coronets" during the shots set in the church. It's used to produce a wonderful, and wonderfully bitter, comic effect.) The song ends as the camera returns to find the young boy swinging on the railing again.