I am slowly building a list of trad songs/tunes that are known both in Britain and Japan.
Some Western folk songs are known actively in Japan, having Japanese lyrics (to the same melody) or being used for English study. Others are simply familiar through movies, TV ads, radio etc. American Folk Revival stuff is pretty well known - Peter Paul and Mary (called PPM), Credence Clearwater Revival (called CCR) and Pete Seeger type stuff.
According to the Nihon Kyoiku Ongaku Kyokai (1934) the first time Japanese formally learned Western music at all was when the British conductor of the Satsuma (Kagoshima) Army Band (a Mr Fenton), taught 30 young band members at Yokohama in 1869. From the Meiji Period Western influence became apparent in schools, not least due to the adoption of Western educational systems. The Ministry of Education certainly promoted some of the well known foreign folk melodies (with Japanese lyrics) via the national curriculum in the 50s and 60s - I am currently going through the text books. Also, a lot of western nursery rhyme tunes are known. Some of these tunes appear in toys, music boxes and ansafones (many manufactured in China!) and beginners' instrument tutors and in the Tokyo Kodomo (Kids) Club popular records. Anyway here are a very few widely known melodies gathered so far. You can add to this list by posting in 'comments'. (J) means it's also a song in Japanese.
Auld Lang Syne (J), Amazing Grace (J), The Water is Wide (TV commercial), Suil A Ruin (J), Greensleeves, Scarborough Fair, Danny Boy, Sally Gardens, Home on the Range, Little Bird (I have heard, what a merry song..), Twinkle twinkle little star (J), The Minstrel Boy, My Grandfathers Clock, I've Been Working on the Railroad, Brahms' Lullaby, Sur le pont d'Avignon, Mockingbird, Pop Goes The Weasel, Happy Birthday To You, When The Saints Go Marching In, Puff The Magic Dragon, Danny Boy, You Are My Sunshine, Greensleeves, Waltzing Matilda, Camptown Races, Sipping Cider Through A Straw, The Blue Bells Of Scotland, Believe Me, If All Those Endearing Young Charms, Annie Laurie, Early One Morning, The Muffin Man, The Man On The Flying Trapeze.See also the tracklists for records produced for kids by Tokyo Kodomo Club 1965-80.
Japan Times Article - multicultural influences on Japanese music/dance/song