These are the songs covered in the Beatles EMI releases. An asterisk indicates more full information elsewhere:
Anna (Go To Him) (Alexander): Arthur Alexander [‘62, Dot 16387]
Chains (Goffin, King): the Cookies [‘62, Dimension 1002]
Boys (Dixon, Farrell): the Shirelles [‘60, Scepter 1211]
Baby It’s You (David, Williams, Bacharach): the Shirelles [‘61, Scepter 1227]
A Taste of Honey (Scott, Marlow): from 1960 Broadway show, this version based on Lenny Welch [‘62, Cadence 1428] *
Twist and Shout (Medley, Russell): Isley Brothers [‘62, Wand 124] *
Till There Was You (Williams): from The Music Man, this version based on Peggy Lee [‘60, Latin ala Lee Capitol ST-1290, also ‘61 U.K. single Capitol CL 15184] *
Please Mister Postman (Dobbin, Garrett, Garman, Brianbert): the Marvelettes [‘61, Tamla 54046]
Roll Over Beethoven (Berry): Chuck Berry [‘56, Chess 1626]
You Really Got a Hold On Me (Robinson): the Miracles [‘62, Tamla 54073]
Devil In Her Heart (Drapkin): the Donays [‘62, Brent 7033]
Money (Bradford, Gordy): Barrett Strong [‘60, Anna 1111] *
Long Tall Sally (Johnson, Penniman, Blackwell): Little Richard [‘56, Specialty 572]
Slow Down (Williams): Larry Williams [‘58, Specialty 626]
Matchbox (Perkins): Carl Perkins [‘57, Sun 261] *
Bad Boy (Williams): Larry Williams [‘59, Specialty 658]
Rock and Roll Music (Berry): Chuck Berry [‘57, Chess 1671] *
Mr. Moonlight (Johnson): Dr. Feelgood & the Interns [‘62, Okeh 7144]; artist formerly recorded as Piano Red
Medley: Kansas City/Hey, Hey, Hey, Hey (Leiber, Stoller / Penniman): Little Richard [‘59, Specialty 664 - a 1955 recording] *
Words of Love (Holly): Buddy Holly [‘57, Coral 61852]
Honey Don’t (Perkins): Carl Perkins [‘56, Sun 234]
Everybody’s Trying to Be My Baby (Perkins): Carl Perkins [‘56, Teen Beat! Sun LP 1225] *
Act Naturally (Morrison, Russell): Buck Owens [‘63, Capitol 4937]
Dizzie Miss Lizzie (Williams): Larry Williams [‘58, Specialty 626]

These are 1961 German recordings, made with Bert Kaempfert for Polydor:
With Tony Sheridan on lead vocals:

My Bonnie (trad., arr. Sheridan): a traditional tune *
When the Saints Go Marching In aka The Saints (trad., arr. Sheridan): a traditional tune *
If You Love [Leave?] Me, Baby aka Take Out Some Insurance On Me Baby (Charlie Singleton, Waldense Hall): Jimmy Reed [‘59, Vee Jay 314]
Sweet Georgia Brown (Bernie, Pinkard, Casey): a 1925 hit for Ben Bernie [Vocalion 15002], Ethel Waters [Columbia 379-D] and Isham Jones [Brunswick 2913]; also Bing Crosby [‘32, Brunswick 6320] and Brother Bones & his Shadows [‘49, Tempo 652, adopted as the theme of the Harlem Globetrotters basketball team]
Nobody’s Child (Cy Coben, Mel Foree): Hank Snow [‘50, RCA 48-0147]
And with John Lennon as vocalist:

Ain’t She Sweet [Ager, Yellen]: 1927 hit for Ben Bernie [Brunswick 3444] and others *

The 1994 release
Live at the BBC is taken from radio broadcasts from 1962 to 1965, and includes many songs not recorded otherwise.
I Got a Woman (Charles): Ray Charles [‘55, Atlantic 1050]
Too Much Monkey Business (Berry): Chuck Berry [‘56, Chess 1635]
Keep Your Hands Off My Baby (Goffin, King): Little Eva [‘62, Dimension 1003]
Young Blood (Leiber, Stoller, Pomus): the Coasters [‘57, Atco 6087]
A Shot of Rhythm and Blues (Thompson): Arthur Alexander [‘62, Dot 16309]
Sure to Fall (In Love With You) (Perkins, Claunch, Cantrell): Carl Perkins [‘56, Sun 235]
Some Other Guy (Leiber, Stoller, Barrett): Richie Barrett [‘62, Atlantic 2142]
That’s All Right (Mama) (Crudup): Arthur Crudup song made famous by Elvis Presley [‘54, Sun 209]
Carol (Berry ): Chuck Berry [‘58, Chess 1700]
Soldier of Love (Cason, Moon): Arthur Alexander [‘62, Dot 16357]
Clarabella (Pingatore): Jodimars [‘56, Capitol F3588], formerly of Bill Haley’s Comets
I’m Gonna Sit Right Down and Cry (Over You) (Thomas, Biggs): made famous by Elvis [‘56, Elvis Presley RCA Victor LPM-1254] *
Crying, Waiting, Hoping (Holly): Buddy Holly [‘59, Coral 62134], one of the tunes recorded on tape in his New York apartment in 1959.
To Know Her Is To Love Her (Spector): the Teddy Bears [‘58, Dore 503]
The Honeymoon Song (Theodorakis, Sansom): ‘An Thimithis to Oniro Mou’ from film The Honeymoon; vocal version by the Italian Marino Marini [‘59]
Johnny B. Goode (Berry): Chuck Berry [‘58, Chess 1691]
Memphis, Tennessee (Berry): Chuck Berry [‘59, Chess 1729] *
Lucille (Collins, Penniman): Little Richard [‘57, Specialty 598]
Sweet Little Sixteen (Berry): Chuck Berry [‘58, Chess 1683]
Lonesome Tears In My Eyes (Burnette, Burnette, Burlison, Mortimer): the Rock ‘n’ Roll Trio (with Dorsey & Johnny Burnette) [‘56, Johnny Burnette & the Rock ‘n’ Roll Trio Coral CRL 57080]
Nothin’ Shakin’ (Fontaine, Calacrai, Lampert, Gluck): Eddie Fontaine [‘58, Argo 5309]
The Hippy Hippy Shake (Romero): Chan Romero [‘59, Del-Fi 4119] *
Glad All Over (Bennett, Tepper, Schroeder): Carl Perkins [‘58, Sun 287]
I Just Don’t Understand (Wilkin, Westberry): Ann-Margret [‘61, RCA 7894]
So How Come (No One Loves Me) (Bryant): the Everly Brothers [’60, A Date With the Everly Brothers Warner Brothers W 1395]
I Forgot to Remember to Forget (Kesler, Feathers): Elvis Presley [‘55, Sun 223]
I Got to Find My Baby (Berry): Chuck Berry [‘60, Chess 1763]
Ooh! My Soul (Penniman): Little Richard [‘58, Specialty 633]
Don’t Ever Change (Goffin, King): the Crickets [‘62, Liberty 55441]

These additional songs appeared on
The Beatles Anthology [1995].
That’ll Be the Day (Allison, Holly, Petty): the Crickets (with Buddy Holly)[‘59, Brunswick 55009]
Hallelujah, I Love Her So (Charles): Ray Charles [‘56, Atlantic 1096]
Searchin’ (Leiber, Stoller): the Coasters [‘57, Atco 6087]
Three Cool Cats (Leiber, Stoller): the Coasters [‘59, Atco 6132]
The Sheik of Araby (Smith, Wheeler, Snyder): 1922 song by the Club Royal Orchestra [Victor 18831] and the Ray Miller Orchestra [Columbia A-3519], and later by Jack Teagarden [‘39, Brunswick 8370].
Besame Mucho (Velasquez, Skylar): 1941 Mexican song by Consuelo Velásquez, a 1944 hit for Jimmy Dorsey [Decca 18574], Andy Russell [Capitol 149] and Abe Lyman [Hit 7072], later by the Coasters [‘60, Atco 6163]
How Do You Do It (Murray): offered first to the Beatles, their arrangement provided Gerry & the Pacemakers with a hit [‘63, Laurie 3162]
Lend Me Your Comb (Twomey, Wise, Wiseman): Carl Perkins [‘58, Sun 287]
Moonlight Bay (Madden, Wenrich): 1912 hit for the American Quartet [Victor 17034] and others, revived by Bing and Gary Crosby [‘51, Decca 27577], charted also for the Drifters [‘58, Atlantic 1187]. It was sung by Alice Faye in the movie Tin Pan Alley [‘40] and Doris Day in On Moonlight Bay [‘51].
Shout (Isley, Isley, Isley): the Isley Brothers [‘59, RCA 7588]
Leave My Kitten Alone (John, Turner, McDougal): Little Willie John [‘59, King 5219]