Here's the flip side of "The Little Fat Policeman" called "The Safety Song" and it too mentions a policeman warning of all the dangers of the street, most particularly, cars. I get a kick out of the line "poor car, poor you"!

The Safety Song

The policeman says, so it must be true,
Bumping into cars isn't good for you.
It hurts the car and it hurts you, too.
Poor car, poor you.
So wait at the corner 'til the red light's gone,
Only a minute when the green light's on,
You can cross the street, and safely, too.

Chorus:
Safe and slow, that's how I go,
There are no lumps and bumps on me, you see,
I watch the light when it's green as grass,
The cars all stop so that I can pass.

The policeman says when you play baseball,
Don't run in the street, no not at all.
A car's so big and you're so small,
Don't chase your ball.
Stay out of the street on your bike and skates,
Cars whiz by and it's way too late,
To cry when you're hurt, you're hurt that's all.

Chorus.