It's slightly oversimplifying things to say Postman Pat is in 4/4.

The first part of it suggests 2/4: there's a very strong stress falling on the "Post" of "Postman" every two beats, and the pace is that of a march.

I'd write it as two bars of 2/4 ("Postman Pat, Postman Pat"). Then one bar of 3/4 ("Postman Pat & his black and white"), then, upon the word "cat" it relaxes into five bars of regular 4/4.

You could, for the sake of writerly simplicity replace the two 2/4s with one 4/4, but it'd be going against the grain of the second "Post". However you look at it, there's unquestionably a bar of 3/4 in there.