Bruce Chanel had had a UK hit with "Hey! Baby!" in the spring of 1962 that had a "similar harmonica", which was played by Delbert McClinton:
http://en.wikipedia.org/wiki/Delbert_McClintonThe Beatles shared a bill with Bruce Chanel and McClinton at the Tower Ballroom, in Wallasey, on 21st JUne 1962. Apart from Lennon liking the harmonica sound, McClinton was born in Lubbock, Texas, which, as the home of Buddy Holly, would surely have intrigued McCartney - assuming he knew about it.
All of this information comes from the book by Ian MacDonald, who detects the influence of Buddy Holly in one part of McCartney's vocal in LOVE ME DO.
MacDonald says LOVE ME DO was recorded three times by The Beatles:
6 June 1962 audition - with Pete Best on drums (on ANTHOLOGY 1)
Note: Lennon sings against his overdubbed harmonica 4 September 1962 - with Ringo on drums
11 September 1962 - with Andy White on drums
Mark Lewisohn (THE COMPLETE BEATLES CHRONICLES) says that the initial pressings of the single used the 4 September recording but later pressings of the single and the PLEASE PLEASE ME album used the 11 September recording (and is thus the standard version).