The common thread about school and songs is the strength of the teacher and flexibility of the curriculum and prevailing culture.In my London kindergarten kids learned "Oranges and LEmons"; "Fly Away Peter, Fly Away Paul" (not known by many Americans apparently); "Ring Around the Roses" (which in England differs from the American version preserving the Black Death connotation); and "goosey goosey gander."
In the London "prep" school (7-13 years of age) in the 60s we were exposed to the typical (?) English folk songs of the middle class: "Lincolnshire Poacher;" "Green Grow the Rushes, Oh;" "London's Burning"; "John Peel"; and all these songs were supplanted by the advent of the modern Gilbert and Sullivan of the 70s: Messrs Rice and Webber (Technicolor Dreamcoat).
Once we went to "public" school (or secondary school), if anyone bothered to sing it was at enforced chapel practice which sounded more like rugby song practice (then again, I suppose rugby songs had their place in the culture too, usually with a covert beer). We did have a peculiar (Devon) boarding school tradition called the "house shout" where we'd do anything from "On Ikley Moor B'aht"; "Widdecombe Fair" to "Never Weather Beaten Sail" (Thomas Campion's 1600's song). And another peculiar instituion was the school song -- some 19th Century anthem to the memory of all who had trod the halls before (and maybe why some felt that they were living in another era).
Coming to the States, well it was Jodies and marching songs ... but that is another thread. And my kids, well they are learning sea chanteys, Arlo Guthrie cowboy renditions of cowboy songs, and "Ging Gang gooly" from their dad ... since music is not a hot subject in their grade schools and sadly (like most things here) must be learned out of school.
Cheers, Brian