We always said - and the joker broke his leg
To the tune of “Comet”
HItler had only one big ball
Tojo had 2 but they were small
Mussolini had one big weenie
And Goebbels had no balls at all
I used deduction to complete this one as I never heard the whole thing.
You are remembering one of Cecil’s columns
Did Hitler have only one testicle?
The bit of doggerel favored by your landlord probably goes something like the following, originally sung by British Tommies during World War II to the tune of the “Colonel Bogey March”:
Hitler has only got one ball,
Goering has two, but very small;
Himmler is very sim’lar,
And Goebbels has no balls at all.
Whistle while you work
Hitler was a jerk
Mussolini bit his weenie
Now it does not work
Speaking of Rudolph, does anyome remember the “western version”? About
Randolph the six gun cowboy
Had a very shiny gun
And if you ever saw it
You would even start to run
…
Santa came to say
'Randolph with your gun so bright
Won’t you shoot my wife tonight?"
Was this like an “official” version? Dr Demento or something? I only learned it from the other kids.
Separate thought:
Does anyone else still default to the parody versions of Christmas songs? 50 years on and I have to stop myself from absent-mindedly singing about “shitty sidewalks”, "in the lane children pissin’ " or “later on we’ll perspire/as we sweat by the fire”.
Childhood combined with a long memory has ruined my Christmas season. 
They often cross my mind. IIRC MAD magazine did an issue with a bunch of them. The only one I partly remember was to the tune of “O Little Town of Bethlehem,” about crime.
With rifles and police dogs,
We guard our homes and stores,
Alive we’ll stay this Christmas Day
If we don’t go outdoors.
Bob Rivers had a few gems.
(To the tune of White Christmas)
I’m screaming at a white sheep dog
Each time he sits upon my chair.
It’s a thing I’m dreading
The way he’s shedding
And coats everything with hair.
I’m screaming at a white sheep dog.
If he should visit you one night
May his bark be worse than his blight
And may all your furniture be white.
I never heard about Alfred breaking his leg, but I’m surprised nobody’s brought up this version yet. This is what I sang:
Jingle Bells, Batman smells
Robin laid an egg
Batmobile lost its wheels
And Batgirl broke her leg
(Chicago suburbs late ‘60s-early ‘70s)
I always understood “and the Joker got away” to be a consequence of the Batmobile losing its wheel. It’s harder for me to see how that lost wheel leads to some third party breaking a leg—it sounds like a non sequitur.
I had never heard any version with anyone breaking their leg (+1 to Thudlow_Boink), but I did hear the both the “Cheerios” and the “Santa smells” versions.
I’ve also heard that version (which kind of makes me wonder if somebody has a throne with some serious hydraulics), but what I heard growing up was this:
Joy to the world, the school burned down,
With all the teachers in it!
They’re looking for the principal,
Whose hanging by the flagpole,
With a rope around his neck,
With a rope around his neck,
With a rope, a rope around his neck!
And here’s another fun Christmas ditty from my childhood:
Deck the halls with gasoline [asterisk]
Fa-la-la-la-la-la-la-la-la!
Strike a match and watch it gleam
Fa-la-la-la-la-la-la-la-la!
Watch the school burn down to ashes
Fa-la-la-la-la-la-la-la-la!
Now aren’t you glad you played with matches
Fa-la-la-la-la-la-la-la-la!
asterisk: One friend sang “kerosene”
Whistle while you work
Nixon is a jerk
Stuck his wiener
in a vacuum cleaner
Now it doesn’t work
We’d replace “Nixon” with our friend’s name.
What are we, 6 years old? Heh.
G.I. Joe
Let one go
All the way to Mexico!
Christmas carol parodies were a pretty regular feature in MAD Magazine.
https://www.madcoversite.com/mad188-carols.html
Commissioner broke his leg (and Robin laid an egg, for that matter), where I grew up. The first time I ever heard the Joker line was when Bart sang it in the Simpsons Christmas Special.
Today was my brother’s birthday, which reminded me of:
Happy Birthday to you!
You belong in a zoo,
You look like a monkey
And smell like one too!
Followed by:
Stand up, stand up
We won’t shut up till you stand up
(Repeat ad nauseum)
To the tune of The Old Gray Mare
Great big globs of
Greasy grimy gopher guts
Mutilated monkey feet
Disconnected horses hooves
French fried eyeballs
floating in a pool of blood
That’s what we always eat.
Randolph the six-gun shooter
Had a very shiny gun
And if you ever saw it
You would turn around and run.
All of the other cowboys
Used to laugh and call him names
They never let poor Randolph
Join in any poker games.
Then one foggy Saturday
The sheriff came to say,
“Randolph, with your gun so bright,
Won’t you shoot my wife tonight?”
…
We sang this one a bit differently:
Great big globs of
Greasy grimy gopher guts
Mutilated monkey meat
Little dirty birdies’ feet
All fried up with all-purpose porpoise pus
You eat it with a knife and spoon
I don’t recall all the stuff after Greasy Grimy Gopher Guts, but the last line was “And I forgot my spoon.”