In the first place, “I’ll Be On My Way” is a Beatles’ composition which they released on their “Live At The BBC” album. However, this is one of the songs that the Fab Four gave away. In this case, it was Billy J Kramer with the Dakotas that took it into the singles charts in 1963.
In Detail
Album
Recorded: 4th April, 1963
Genre: Pop
Track Duration: 1:58
Record Label: Apple Records
Songwriters: Lennon:McCartney
Producers: Bryant Marriott, George Martin (Executive Producer)
Performers
John Lennon: vocals, acoustic guitar
Paul McCartney: harmony vocals, bass
George Harrison: lead guitar
Ringo Starr: drums

Live At The BBC Album (1994)
Track Source
I’ll Be On My Way
On the 26 April, 1963, Billy J Kramer with the Dakotas released the single, “Do You Want to Know a Secret” which is a Lennon-McCartney song. On the B-side is another song from the prolific songwriters, “I’ll Be On My Way”. Billy J Kramer with the Dakotas took this single to number 2 in the UK charts. It didn’t reach the top spot because The Beatles themselves were there with, “From Me to You“.
Kramer was from Bootle, near Liverpool. Not only that but he and his group had the same manager as The Beatles, Brian Epstein. They also recorded their song at the Abbey Road studios under producer, George Martin. So, it’s not surprising then that this song was successful for the band. In fact, it was Kramer’s first single and also appears on the 1979 album, “The Songs Lennon and McCartney Gave Away”.
It certainly sounds like influence for the song came from Buddy Holly and they would perform it at some of their early gigs. Be that as it may, the Fab Four only recorded the song once and that was for BBC Radio.
Recording Studio
The Fab Four recorded their one and only version of “I’ll Be On My Way” on the 4th April, 1963, at the BBC Paris Theatre in London. This was for the radio show, “Side by Side” airing on the 24th June, 1963. This version is now available on The Beatles’ 1994 album, “Live At The BBC”.