I've remembered the first lines of the verse which goes something like this:
Take me back to Manchester where it's raining
I want to wash my feet in Albert Square
I'm all agog, for a good thick fog
I'm only really happy when it's raining cats and dogs
I want to wash my feet in the Irwell
.........................
I think it was probably one of those little ditties which people made up their own verses
I doubt very much if it was ever written down so I'll have to rely on people's memeories.