Here are four I use on car rides.
Old McDonald works well to Sweet Georgia Brown.
Old McDonald had a farm. E-i-e-i-o.
And on that farm he had a (animal). E-i-e-i-o.
With a (noise) (noise). [Softly, with jazz hands: Here and there!]
And a (noise) (noise). [Softly, with jazz hands: Everywhere!]
Old McDonald had a farm. E-i-e-i-o. [bum bum bum]
The Alphabet Song works well to Stray Cat Strut almost without change.
The Wheels on the Bus fits It's Only a Paper Moon so well I need not comment.
There Was an Old Lady Who Swallowed a Fly fits It Had to be You. Just add some "da da ta da da" after the first two lines.
What others can you think of?