The Beatles recorded a version of the Elvis Presley song, “That’s All Right (Mama)” and it is available on their “Live At The BBC” album from 1994. Although The Beatles would perform many Elvis songs, they only ever recorded three of them. This version is from their “Pop Go The Beatles” radio show in 1963.
In Detail
Album
Recorded: 2nd July, 1963
Studio: BBC’s Maida Vale Studios, London
Genre: Rockabilly
Radio Show: Pop Go The Beatles, 16th July, 1963
Track Duration: 2:56
Record Label: Apple
Songwriter: Arthur Crudup**
Producer: Terry Henebery
** also the original singer of his own song.
Performers
Paul McCartney: vocals, also the bass guitar
John Lennon: rhythm guitar
George Harrison: lead guitar
Ringo Starr: drums

Live At The BBC Album (1994)
Track Source
That’s All Right (Mama)
Although The Beatles recorded this song, it is more famous for being Elvis Presley’s debut single in 1954. However, the singer/songwriter, Arthur ‘Big Boy’ Crudup, released his own single version as far back as 1947. So, it’s clear that this song has a pedigree going back many years.
In their early days, The Beatles, like most up and coming groups, would cover old songs such as this one. Although The Beatles covered a lot of Elvis Presley songs, they only recorded three of them at the BBC for airing on radio shows.
The Beatles’ version of “That’s All Right Mama” that is commercially available to us right now is on their 1994 album, “Live At The BBC”. They recorded this song on the 2nd July, 1963, at the BBC sound studios in Delaware Road, Maida Vale, London. Then, on the 16th July, 1963, the Fab Four fans heard it live on the BBC’s “Pop Go The Beatles” radio show.
While Paul McCartney sang the lead vocals, the band performed the song to closely match the original rockabilly style by Elvis.