The SEGA Master System version has the most accurate music order as every song gets to be heard in each of the 4 modes and a part from 4 music tracks the rest are repeated twice in each mode.
From what i've personally discovered and learned about the Lemmings songs*
Levels 1 & 18 Can Can (Orpheus In The Underworld) composed by Offenbach
Levels 2 & 19 Pachelbel's Canon In D Major composed by Johann Pachelbel
Levels 3 & 20 (Tailor Made For Blockers) unknown composer
Levels 4 & 21 (Miners And Climbers) unknown composer
Levels 5 & 22 Dance Of The Four Little Swans composed by Tchaikovsky
Levels 6 & 23 Puggs In Space Demo Track By Tim Wright
Levels 7 & 24 (The Ascending Pillar Scenario Disco-Been There, Seen It, Done It,) unknown composer
Levels 8 & 25 How Much Is That Doggie In The Window composed by Bob Merrill in 1952 sung by Patti Page
Levels 9 & 26 Dance Of The Reed Flutes (from the Nutcracker Suite) composed by Tchaikovsky
Levels 10 & 27 (Smile If You Love Lemmings) unknown composer
Levels 11 & 28 Rondo Alla Turca composed by Mozart
Levels 12 & 29 London Bridge Is Falling Down (first published in 1744) unknown composer
Levels 13 & 30 (We All Fall Down) composer unidentified although it may be an original composition by Matt Furniss 1992
Level 14 Miniature Overture (also from the Nutcracker Suite) by Tchaikovsky
Level 15 Scotland The Brave unknown composer
Level 16 She'll Be Coming Round The Mountain - song is originally american and was first sung in the late 1800's by railworker gangs, it is a derivation of a Negro spiritual known as "When the Chariot Comes". unknown composer
Level 17 My Old Man Said Follow The Van composed by Fred W. Leigh and Charles Collins song made popular by Marie Lloyd.
Lemmings Title Theme. unknown composer although it may be an original composition by Matt Furniss 1992
Other Songs On The Mega Drive Version:
Level 18 Sunsoft - Sunsoft Special = Overworld, Hebereke / Ufouria Theme Composed By Naoki Kodaka, Nobuyuki Hara, Shinichi Seya 1991
Mega Drive Ending Theme - March Of The Mods composed by Tony Carr
Happy now finlay? No spelling errors.
I'll just add that if Tim Wright and Brian Johnston did compose the ''unknown composer'' tracks, then they're not credited in the SEGA Master System and SEGA Mega Drive versions of Lemmings which is the two versions I personally grew up on so thats why I marked them as unknown. And I believe the Doggie song was out of copyright for a while as old songs tend to go out of copyright after 50 years or so but someone or some music publishing company bought the rights to the song again recently.
And since you like to nitpick on everything I post i'll throw one back at you. It's neither Johnson or Johnstone, it's (Johnston).