Origin of the whistling sound when things fall

Because in WW2, the Germans attached actual whistles to bombs as a psychological tactic.

When you are watching a Hollywood reenactment of a famous World War 2 battle, the whistling cries of falling bombs certainly help to build tension, but there is some truth behind this terrifying sound. During World War II, the Germans designed their bombs with a special whistle that would make that screaming cry as they fell towards the cities below. Typically, a metal object with sharp edges, like a bomb would make a small noise “hissing” sound as it fell through the air, unless it happened to exceed the sound barrier (343 meters per second), which would result in a sonic boom.

However, by designing bombs that had a whistle-like attachment, the Germans were utilizing psychological warfare as much as physical warfare. The whistle became associated with death from above, and since the bombing raids in London and other European cities were often done in the middle of the night, that wailing cry became a nightmare-inducing and anxiety-striking sound. Survivors of the London bombing still remember those haunting whistles of death from their underground shelters.

Although I personally find @Paulie_D 's answer to be satisfactory regarding the historical origins of the noise, I wanted to add another response specifically relating to the noise's introduction to cartoons.

According to Mark Mangini, who was a sound designer for Hannah Barbera back in the day:

Music and sound effects had to be performed at the same time in the same space. Musical instruments were used to make the effects because they were easy to find, and easy to manipulate. In this Tom and Jerry clip, the sound of a frying pan hitting Tom’s face is played by a cymbal crash. (cymbals)

Specifically relating to the slide-whistle noise for a falling item or bomb:

The percussionist would probably have it as part of their kit, and it was just natural to convey going up (slide up) or down (slide down). You could manipulate them in any one of a number of ways, very quickly or very slowly.

Source: https://99percentinvisible.org/episode/classic-cartoon-sound-effects/transcript/

I would highly recommend listening to the full interview, as it's a fascinating topic.

The knowledge predates WW2 - in 1934 the book Biggles of the Camel Squadron was written by WE Johns, and describes a scene in WW1, thus:

“I wish you wouldn’t interrupt!” snarled Henry. “When I was at Thetford, a fool came over from Narborough, on Christmas morning, and dropped an empty bottle from about ten thousand feet. We didn’t know it was a bottle. We thought it was just the sky falling down. At first it whistled, then it shrieked, and then it——” Henry threw up his hands in a despairing gesture. “The din was like nothing on earth. It made more noise than a score of 230-pound bombs. Now, my point is this: If one bottle can do that, think of the noise two or three dozen bottles would make falling at once! I’ll bet the gunners would stick their heads in their dugouts when that lot started warbling. They’d go to earth like a lot of rabbits with a terrier around.”

So this suggests that in the Great War falling bombs made whistling noises.

Details and full text https://www.fadedpage.com/showbook.php?pid=20200803