It turns out what I had gleaned from who-knows-where was only the chorus of a long song titled "The Hippopotamus," by Flanders and Swann, which tune and words form the soundtrack of a suitably watery video.
One can find the words of the verses online, but they aren't really for children. The chorus alone was sufficient to spark up our family's muddy excursions, and it goes like this:
Mud, mud, glorious mud,
Nothing quite like it for cooling the blood.
So follow me, follow,
Down to the hollow,
And there let us wallow
In glorious mud!