Sorry to disappoint whoever it was praised us brits. for holding out as we did - but we weren't alone; the Russians did an even greater job of holding out, and paid a far greater price (ok, so millions more of their citizens also paid a price after the war, but that's another issue).
Generally I dislike the "re-write history guessing games" but it frightens me to think what would have happened had Germany not opened her second front against Russia.
As for the original question . . . at the age of 5 (which would make it about 1953) I had my tonsils removed. On my return from hospital awaiting me was a wind-up gramophone, a present from my Grandfather (the last wealthy member of our family, but that's several other stories!!). Of the three 78s he sent with it one of them had "Run, Rabbit Run" on one side, and "Hang out the washing" on the other. I'd guess these were the Flannagan and Allan recordings.
I'd also guess (no evidence at all) that these were amongst the very few "peoples" songs (as opposed to "art songs, which is where I'd place "Bluebirds") from WWII to become popular.