Shep Woolley recorded this on his first LP Pipe Down, under the title "The Best of Society" [as per chorus posted earlier], and may be influenced by / borrowed from Mike Harding's.
Some verses not yet listed, this first making a sensible 'intro' to the other, less connected, ones:

Now, our part of town is a proper disgrace.
People flock in to see it from all over the place.
For miles, on a Sunday, you should see them all come: [I'd edit that to 'From miles away, on Sundays… ']
they're pulling it down next week and building a slum.

Me mum sent me to the butcher's to buy a sheep's head
and when I got there, I told him what she'd said.
I said: "Have you got a sheep's head?" and he gave me a stare;
he said: "No, mate, I bloody haven't: it's the way I part me hair."

Also [which, being topical then, SW may have written]:

I went to the airport, and there was Barb'ra Castle*.
What, with reporters and photographers, it was a bit of a hassle.
She said: "I'm off to Bangkok"; I thought "What a to-do. [I reckon 'I said: "Bully for you!… ' makes better sense]
You've banned ev'rything else, now you're banning that, too."

[these last don't scan as well, but can be rendered correctly]

* a high-profile politician at the time

The song then continues with a different, widely-used, tune, with at least this verse, and prob. others I've forgotten:

The black cat piddled in the white cat's eye.
The white cat said: "Cor, blimey!"
So the black cat said: "It's yer own bloody fault:
you shouldn've stood behind me."

I can't recall if his version contains the 'tomato' one: I think it does. He def. sings the Miss Brown & the soap [surprisingly, given the context: a live recording to a largely Navy-based audience, with some pretty un-PC racist material – SW was a Naval gunnery tutor – it's a "came down mighty quick" clean variant!], and the Mary the milkmaid & the cow jumping up and down variants. There may be others, as I've subsequently heard it sung by others, in most of its variety [incl. the "Hold your row" chorus, which I think is northern English, poss, Geordie], and they've got conflated in my mind.