I heard this many years ago at a house concert by Roy Harris. He used it as a kind of "Get your attention song" to open a concert. When he stopped traveling over here, I adopted it.
The only title I know it as is the same as the chorus:
It is half sung, half bellowed

"There was an old woman, she kept fat hogs,
She made plum puddings-they poisoned the dogs,
    They were black, white, yellow and green.

Cho: "Black, white, yellow and green,
      The rarest plum puddings you ever have seen.
      The were black, white, yellow and green.

She took them a threw them into a pot,
And boiled them till they were steaming hot,
    They were black, white, yellow and green.

Cho:"Black, white, yellow and green,
      The rarest plum puddings you ever have seen.
      They were black, while, yellow and green.

She baked them till they were blazing red,
The one was like leather, the other like lead.
   They were black, white, yellow and green.
Cho.

She took out a pin and pricked their skins,
The gravy ran out and the maggots ran in.
    They were black, white, yellow and green.

Cho:

   She took them a put them down on the floor,
   Each took to its feet and ran out of the door.
    They were black, white, yellow and green.

Cho:

   She took them and put them up on a shelf...
   If you want any more, you can sing it yourself! (or "help yourself)
    They were black, white, yellow and green.

Cho:Black, white, yellow and green,
      The rarest plum puddings you ever have seen.
      The were black, white, yellow and greeeeenn!
   
Here is a link, but I'll guarantee the tune Roy used was not that 'tinkley'. https://www.8notes.com/scores/3605.asp