The further I go back in time, the fewer songs I recognise. Because a whole stack of Time Kicks Back is set in Bletchley Park during 1940, I’m gonna spend a wee while looking at that decade. I couldn’t find enough songs I knew/liked to do more than a single summary for the entire decade.
Here they are:

1. That Old Black Magic – Judy Garland
2. Pennsylvania 65000 – The Andrews Sisters
3. White Christmas – Bing Crosby
4. Nature Boy – Nat King Cole
5. We’ll Meet Again – Vera Lynn
6. Deep In The Heart Of Texas – Bing Crosby
7. Pistol Packin’ Mama – Bing Crosby & The Andrews Sisters
8. These Foolish Things – Frank Sinatra
9. Over The Rainbow – Judy Garland
10. (There’ll Be Blue Birds Over) The White Cliffs Of Dover – Vera Lynn
11. Swanee – Al Jolson
12. Whispering Grass – Ink Spots
13. The Trolley Song – Judy Garland
And a few bubbling under:
You Are My Sunshine – Bing Crosby
Lili Marlene – Anne Shelton
Begin the Beguine – The Andrews Sisters
If I Had My Way – Bing Crosby
Too-Ra-Loo-Ra-Loo-Ra! (That’s An Irish Lullaby) – Bing Crosby
Don’t Fence Me in – Bing Crosby & the Andrews Sisters
Ac-Cent-Tchu-Ate the Positive – Bing Crosby & the Andrews Sisters
Meet Me in St. Louis – Judy Garland
Saturday Night (is the Loneliest Night in the Week) – Frank Sinatra
Moonlight Becomes You – Bing Crosby
Nancy (With the Laughing Face) – Frank Sinatra