From Wikipedia: The opening music and "Corn Rigs" are arrangements of the Robert Burns ballads "The Highland Widow's Lament" and "Rigs O' Barley", respectively. The instrumental parts of the score are based on traditional Scottish, Irish, and English tunes such as "Miri it is", the strathspey "Robertson's Rant" jig, and "Drowsy Maggie" reel. "Chop Chop" is based on the nursery rhyme "Oranges and Lemons". "Procession" is an arrangement of the tune of the Child Ballad "Fause Foodrage". Although some of the music is Scottish, and the film is set in the Hebrides, no traditional Scottish Gaelic numbers are featured.
Giovanni “had 6 weeks to research, prepare, compose and record the soundtrack, all of which had to be done before the first day’s shooting.” - Cinefantastique, Vol 6, #3 (1977) He had written 8 or 9 songs but some had to be cut as Shaffer felt it slowed the plot down to be having music "every 20 seconds." (Brown)
The only London Symphony Orchestra involvement was "Maurice Murphy playing the very long ramshorn note outside the cave (but on a trumpet of course) - and the closing credits." Gary Carpenter, 12/2/15, Facebook Wicker Man group.
From Gary's website: recordings were done at De Lane Lea, Pye and Shepperton, excluding the on-set recordings of The Tinker of Rye and The Landlord's Daughter's vocals.
See also audio recordings
Music in each version
Individual songs and other music used in the film
Dream sequence (cut from all versions)
The best review of the music I've seen here.
Songbook available here.
See these two BFI videos: Gothic - Sing Cuckoo, The Story of the Wicker Man soundtrack, and Folk Musicians discuss the Wicker Man.